Algebraic Quantum Statistical Mechanics

BCAM Course 2010/2011

Lecturer: Jean-Bernard Bru

Dates: 4 October 2010 – 23 February 2011 (36 lectures of one hour each)

Abstract

The aim of the course is to give mathematical bases of the theory of phase transitionsin quantum statistical mechanics. Phase transitions are studied via equilibrium statesof a given quantum mechanical system, which can be mathematically defined in differ-ent ways: for instance, as minimizers of the free-energy density, as tangent functionalsto convex functions (pressure) on a Banach space of interactions, or as KMS (Kubo-Martin-Schwinger) states. From the mathematical point of view, the existence of a phasetransition for a quantum mechanical system can be seen via the transition to one equilib-rium state to several ones at a specific temperature. Since von R. Haag’s observation inthe fifties, it is known that the existence of several equilibrium states implies several (notequivalent) representations of the observable algebra of the system. This remark moti-vates the introduction an algebraic formulation of quantum statistical mechanics withoutany (explicit) references to Hilbert spaces. This description will be the main subject ofthis course on the mathematics of quantum phase transitions.

Bibliography:

[1] Emch, Algebraic Methods in Statistical Mechanics and Quantum Field Theory.Dover. New York, 2000.

[2] Brattelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechan-ics, vols. I and II. 2nd ed. Springer. New York, 1996.

[3] H. Araki and H. Moriya, Equilibrium Statistical Mechanics of Fermion lattice Sys-tems. Rev. Math. Phys. 15 (2003) 93–198.

[4] B. Israel, Convexity in the theory of lattice gases. Princeton Series in Physics.Princeton Univ. Press, 1979.

1

.

Based on the monograph done by J.-B. Bru and Walter de Siqueira Pedra

University of the Basque Country and Johannes Gutenberg–University of Mainz

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Contents

1 Mathematical Foundations 5

1.1 Mathematical description of physical systems . . . . . . . . . . . . . . . . . . . . 5

1.2 C∗–algebra and its set of states as a model for Axioms 1–11 . . . . . . . . . . . 15

1.2.1 C∗–algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.2.2 States on C∗–algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.3 GNS (Gelfand-Neumark-Segal) representation of states . . . . . . . . . . 22

1.2.4 Commutative C∗–algebras and dispersion–free states . . . . . . . . . . . 24

1.2.5 Conclusion: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 Classical systems versus quantum ones . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.1 Classical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

1.3.2 Quantum systems and Heisenberg uncertainty principle . . . . . . . . . . 27

1.4 C∗–dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4.1 Heisenberg picture of quantum mechanics (QM) . . . . . . . . . . . . . . 28

1.4.2 Schrodinger picture of quantum mechanics . . . . . . . . . . . . . . . . . 30

2 Equilibrium State of Finite Systems 34

2.1 Finite systems and Gibbs state . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.2 Gibbs state as minimizer of the free–energy – Maximum entropy principle . . . . 35

2.3 Gibbs states as tangent functionals . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Infinite Lattice Systems 40

3.1 Spin and fermion algebras on lattices . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.1 Spin algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1.2 Fermion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 The set E1 of translation invariant (t.i.) states . . . . . . . . . . . . . . . . . . . 43

3.3 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.4 Energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.5 Entropy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4 Equilibrium States of Infinite Systems 52

4.1 Equilibrium states as minimizers of the free–energy density . . . . . . . . . . . . 52

4.2 Gibbs states versus equilibrium states . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Equilibrium states as tangent functionals . . . . . . . . . . . . . . . . . . . . . . 56

4.3.1 The set E1 as a subset of the dual space W∗1 . . . . . . . . . . . . . . . . 56

4.3.2 The pressure as the Legendre–Fenchel transform of the free–energy density 583

5 Strong Coupling BCS–Hubbard Hamiltonian and Superconductivity 59

5.1 Local model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Superconductivity via long range interactions . . . . . . . . . . . . . . . . . . . 60

5.3 The approximating Hamiltonian method . . . . . . . . . . . . . . . . . . . . . . 61

5.4 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.5 Equilibrium States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6 Appendix 66

6.1 Locally convex spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Lower–semi–continuous functionals . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.3 Compact convex sets and Choquet simplices . . . . . . . . . . . . . . . . . . . . 73

6.4 The Legendre–Fenchel transform . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.5 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.5.2 Spectrum in algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.5.3 C∗–Homomorphism and representation of C∗–algebras: . . . . . . . . . . 83

6.5.4 The positive elements of a C∗–algebra: . . . . . . . . . . . . . . . . . . . 84

6.5.5 Entropy and conditional expectations in C∗–algebras . . . . . . . . . . . 86

6.6 Additional details on Gibbs state . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.6.1 Gibbs state as KMS (Kubo-Martin-Schwinger) states . . . . . . . . . . . 96

6.6.2 Gibbs state and zero–law of thermodynamics . . . . . . . . . . . . . . . . 98

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1 Mathematical Foundations

1.1 Mathematical description of physical systems

Definition 1.1 (Observables)A physical system S is described by its physical properties, i.e., by a non–empty set O = OS = ∅of all physical quantifies, which can be measured in S, as well as by their relations betweenthem. Elements A ∈ O are seen as operators and are called observables. They represents theapparatus1 and the domain WA ⊂ R of all physical quantifies for A ∈ O.

Axiom 1For all A ∈ O, its measure belongs to a bounded real set WA ⊂ R.

We give now elementary properties satisfied by observables, which correspond to the struc-tural axioms 2 and 3:

Axiom 2If A ∈ O and P is any polynomial with real coefficients, then there is an observable P(A) ∈ O.If P1 and P2 are two polynomials, both with real coefficients, then for all A ∈ O,

P1(P2(A)) = P1 P2(A),

where P1 P2 is the polynomial defined for x ∈ R by

P1 P2(x) = (P1(P2(x)).

Definition 1.2(i) For any A ∈ O and all λ ∈ R, λA := P(A) with P(x) = λx.(ii) For any A ∈ O and all n ∈ N0, An := P(A) with P(x) = xn.

The element λA ∈ Θ described here the physical quantify A ∈ Θ rescaled by λ ∈ R, whichcould be seen as a change of unit. The polynomial P(A) corresponds to the measure of afunction P(·) of the physical quantify A. This motivates the following axiom:

Axiom 3For all A ∈ O and all polynomial P with real coefficients,

WP(A) = P(WA) := P(x) : x ∈ WA.

Lemma 1.3 Assume Axioms 2 and 3. Then, for all λ ∈ R, there is an observable λ ∈ O suchthat Wλ = λ.

Proof: It suffices to choose λ : = P(A), for any A ∈ O and P(x) = λ.

Note that, a priori, the set Wλ = λ may defines several operators.

1Measuring device

5

Definition 1.4 (States of a physical system)A state ρ is a map from O into R and represents the statistical distribution of all measures ofany observable A ∈ O. The set of all possible states is denoted by E.

Then, ρ(A) ∈ R is the expectation value of the measure of the physical quantify A ∈ O whenthe system S is in the state ρ ∈ E.

Definition 1.5 (Dispersion–free states)A state ρ ∈ E is dispersion–free with respect to A ∈ O if the statistical distribution of allmeasures of A is concentrated on a unique value a ∈ R, i.e.,

ρ(A) = a ∈ WA and ρ(P(A)) = P(a)

for all polynomials P with real coefficients. The set of all dispersion–free states with respect toA ∈ O is denoted by EA.

Remark 1.6From Axioms 2 and 3 it follows that EA ⊂ EP(A) for all polynomials P and all observablesA ∈ O.

Note that one obviously has ρ (λ) = λ for all λ ∈ R and ρ ∈ E. This motivates the nextaxiom:

Axiom 4(i) For all A ∈ O and ρ ∈ E,

inf WA ≤ ρ(A) ≤ sup WA.

(ii) For all A ∈ O, ρ ∈ E and λ ∈ R: ρ(λA) = λ ρ(A).

Lemma 1.7Assume Axioms 2, 3 and 4. Then, for all λ ∈ O such that Wλ = λ with λ ∈ R (cf. Lemma1.3), Eλ = E, i.e., any state ρ ∈ E is dispersion–free w.r.t. the observable λ ∈ O.

Definition 1.8 (Spectrum of observables)The spectrum of all A ∈ O is defined by

σ (A) :=ρ (A) : ρ ∈ EA

⊂ R.

Lemma 1.9Assume Axioms 2, 3 and 4. For all A ∈ O, σ(A) ⊂ WA and for all polynomials P with realcoefficients

P(σ(A)) ⊂ σ(P(A)) ⊂ P(WA).

Proof: See Remark 1.6.

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Axiom 5 (Spectrum as possible measures)For all A ∈ O, σ(A) = WA.

In other words, the spectrum of any A ∈ O is, per assumption, all the possible measure ofthe physical quantify A.

Definition 1.10 (Order on O)We say that A ≥ B for A,B ∈ O if ρ (A) ≥ ρ (B) for all ρ ∈ E.

Note that the order ≥ is reflexive, i.e., for all A ∈ O, A ≥ A, and transitive, i.e., A ≥ B andB ≥ C imply A ≥ C. Its antisymmetry is not clear and it is assumed in the next axiom:

Axiom 6 (States separate observables)The order is antisymmetric: A ≥ B and B ≥ A imply A = B, i.e., ρ (A) = ρ (B), for allρ ∈ E, yields A = B. In particular, the order ≥ defines a partial order in O.

Axiom 6 means that states separate observables.

Lemma 1.11 (Uniqueness of the observable λ)Assume Axioms 2, 3, 4 and 6. Then, for all λ ∈ R, the set Wλ := λ defines a uniqueobservable λ ∈ O.

Proof: Clear.

Remark 1.12 (Uniqueness of the observable λA)Axioms 4 and 6 implies that, for all A ∈ O, λA ∈ O is the unique observable such thatρ(λA) = λρ(A) for all ρ ∈ E.

Therefore, the set O includes the observables 0 and 1 which are the unique observables whichcorrespond to the measuring devices which always measure 0 (W0 = 0) and 1 (W1 = 1),respectively.

Definition 1.13 (Positive observables)An observable A ∈ O is said to be positive when A ≥ 0.

Lemma 1.14Assume Axioms 2–6. Then, A ∈ O is positive if and only if (iff) WA ⊂ R+

0 .

Proof: Clear.

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Axiom 7 (Addition of observables)For all A,B ∈ O, there is an observable A + B ∈ O such that ρ (A+B) = ρ (A) + ρ (B) forall ρ ∈ E. Additionally, for all polynomials P1,P2 with real coefficients

(P1 + P2)(A) = P1(A) + P2(A)

with the polynomial (P1 + P2) defined, for x ∈ R, by

(P1 + P2)(x) = P1(x) + P2(x).

Remark 1.15(i) Due to Axioms 6, A+B ∈ O is unique.(ii) Since ρ (A) = ρ (A) + ρ (0) = ρ (A+ 0) for all ρ ∈ E,

A = A+ 0 = 0+ A

for all A ∈ O.

Theorem 1.16 (Vector space structure)Assume Axioms 2–7. Then, (O,+, ·) is a real vector space.

Proof: (O,+) is an abelian group with neutral element 0. Indeed, for all A,B ∈ O, A + B =B + A results from Axioms 7 and Remark 1.15 (i) and 0 is the neutral element because ofRemark 1.15 (ii). The associativity of the addition, i.e., (A + B) + C = A + (B + C), is alsotrivial. For all A ∈ O, its inverse is −A := (−1)A because by Axiom 4 and Remark 1.15 (i) as

ρ ((−1)A+ A) = −ρ (A) + ρ (A) = 0 = ρ (0) ,

for all ρ ∈ E. Additionally, the distributivity, i.e., the equality λ (A+B) = λA + λB for allA,B ∈ O and λ ∈ R, is also trivial.

Definition 1.17 (Norm of observables)For all A ∈ O,

A = supρ∈E

|ρ (A)| .

Remark 1.18Axioms 1 and 4 implies that, for all A ∈ O, A ≤ sup

x∈WA

|x| < ∞ whereas Axiom 5 yields

A = supx∈WA

|x| = supx∈σ(A)

|x| .

Theorem 1.19Assume Axioms 1–7. Then, (O, · ,+, ·) is a normed vector space.

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Proof: As |ρ (λA)| = |λ| |ρ (A)| we have λA = |λ| A for all A ∈ O and λ ∈ R. Moreover,if A = 0 then ρ (A) = ρ (0), for all ρ ∈ E, which in turn implies A = 0 by Axiom 6. Thetriangle inequality is also trivial to verify:

A+B ≤ supρ∈E

(|ρ (A) + ρ (B)|)

≤ supρ∈E

|ρ (A)|+ supρ∈E

|ρ (B)|

= A+ B .

Theorem 1.20 (C∗–property of the norm)Assume Axioms 1–7. Then, for all A ∈ O, A2 = A2.

Proof: From Definition 1.17,WA ∪W−A ⊂ [−A, A].

Let B := A21− A2 and P be the polynomial defined for any x ∈ R by

P(x) := (A − x)(A+ x).

From Axiom 7, B = P(A). Using Axiom 3,

WB = WP(A) = P(WA) ⊂ P([−A, A]) ⊂ R+0 .

From Lemma 1.14, B ≥ 0. In particular,

A2 = ρ(A21) ≥ ρ(A2) ≥ 0,

for all ρ ∈ E, i.e., A2 ≥ A2.Let

B := A21± 2AA+ A2

and P be now the polynomial defined for any x ∈ R by

P(x) := (x± A)2.

In the same way,B = P (A) = A2 1+A2 ± 2 AA ≥ 0

which means that2A|ρ(A)| ≤ A2 + ρ(A2) ≤ A2 + A2

for all ρ ∈ E, i.e., A2 ≤ A2.

Axiom 8The normed vector space (O, · ) is complete.

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Therefore, from Axioms 1–8 combined with Theorem 1.19, we have that O is a Banach space.In particular, its dual space is well–defined:

Definition 1.21 (Dual space of O)O∗ is the set of all continuous R–linear functionals from O to R.

Remark 1.22A linear map ρ from O to R is continuous iff

ρ := supA∈O,A≤1

|ρ (A)| < ∞.

Definition 1.23(i) ρ ∈ O∗ is positive if ρ (A) ≥ 0 for all A ≥ 0, A ∈ O. The set of all positive continuousR–linear functionals is denoted by O∗

+.(ii) ρ ∈ O∗ is normalized if ρ(1) = 1. The set of all normalized continuous R–linear functionalsis denoted by O∗

1.(iii) O∗

+,1 := O∗+ ∩O∗

1 is the set of all positive normalized continuous R–linear functionals.

Axiom 9 (Observables separate states)Let ρ, ρ ∈ E. If ρ (A) = ρ (A), for all A ∈ O, then ρ = ρ.

Remark 1.24If Axiom 9 holds, then we can identify a state ρ ∈ E with the expectation family ρ (A)

A∈O,i.e., with a unique functional from O to R.

Lemma 1.25Assume Axioms 1–9. Then, E ⊆ O∗

+,1, i.e., states ρ ∈ E are positive normalized continuousR–linear functionals.

Proof: Clear.

Axiom 10By assumption, E = O∗

+,1.

Remark 1.26 (E as a convex set)O∗

+,1 is a convex set, that is, for any λ ∈ [0, 1] and ρ, ρ ∈ O∗+,1,

λρ+ (1− λ) ρ ∈ O∗+,1.

Therefore, by Axiom 10, E is a convex set.

Definition 1.27 (Pure and mixed states)(i) Pure states. Extreme point of the convex set E = O∗

+,1 are states ρ ∈ E such that if

ρ = λρ+ (1− λ) ρ

for λ ∈ (0, 1) then ρ = ρ = ρ. Extreme states are also called pure states.(ii) Mixed states. Non–extreme points are called mixed states.

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For all ρ ∈ E, A ∈ O and ε > 0 we define

Uρ

(A,ε) := ρ ∈ E : |ρ (A)− ρ (A)| < ε .

For any ρ ∈ E, all n ∈ N, A1, . . . , An ∈ O, and ε1, . . . , εn > 0, let

Uρ

(A1,ε1),...,(An,εn):=

n

j=1

Uρ

(Aj ,εj).

Definition 1.28 (Physical topology)The physical topology τph ⊂ 2E of E is the topology constructed from the family

Uρ

(A1,ε1),...,(An,εn): n ∈ N, A1, . . . , An ∈ O, ε1, . . . , εn > 0

.

Remark 1.29 (Weak∗–topology)If Axioms 1–9 holds, i.e., O is a Banach space and E ⊆ O∗

+,1, τph is nothing else than therelative topology of E seen as a subset of the topological space (O∗, τ ∗), where τ ∗ is the weak∗–topology of O∗.

Lemma 1.30 (Compacticity of E)Assume Axioms 1–10. Then E is convex and compact with respect to the physical topology τph.

Proof: E is convex by Remark 1.26. From Banach–Alaoglu’s theorem, the unit ball

B1 := ∈ O∗ : ≤ 1

is weak∗–compact. If O∗+,1 is a closed subset of B1, then O∗

+,1 is also weak∗–compact. Theassertion then follows because of Axiom 10 (E = O∗

+,1) and Remark 1.29 (τph = τ ∗).

Closedness of O∗+,1 is proven as follows: From Definition 1.23, O∗

+,1 = O∗1 ∩O∗

+ with

O∗1 = ρ ∈ O∗ : ρ(1) = 1,

andO∗

+ =

A∈O, A≥0

ρ ∈ O∗ : ρ(A) ≥ 0.

The subsets 1 and [0,∞) in R are closed and the map ρ → ρ(A) ∈ R is continuous w.r.t. theweak∗–topology for any A ∈ O, by Corollary 6.26 (ii). Then, O∗

1 and O∗+ are weak∗–closed.

Theorem 1.31 (Krein–Milman)Let K ⊂ X be any (non–empty) compact convex subset of a locally convex topological vectorspace X . Then, one has:(i) The set E(K) of its extreme points is non–empty.(ii) The set K is the closed convex hull of E(K), that is,

K = co (E(K))

with

co (M) = λ1x1 + · · ·+ λnxn : λ1 + · · ·+ λn = 1,

x1, · · · , xn ∈ M .11

Proof: See W. Rudin, Functional Analysis, page 75.

Corollary 1.32 Assume Axioms 1–10. Then, any state of the (non–empty) set E can be given,up to arbitrarily small errors in the sense of τph, by convex sums of pure states. This meansthat, for any ρ ∈ E, n ∈ N, A1, . . . , An ∈ O, ε1, . . . , εn > 0, there are λ1, . . . ,λk ≥ 0 such thatλ1 + · · ·+ λk = 1, and a family

ρj

k

j=1of pure states ρ

j∈ E(E) such that

k

j=1

λj ρj ∈ Uρ

(A1,ε1),...,(An,εn).

Definition 1.33 (Symmetric product)Under Axioms 2–7 we define the operation • from O ×O to O by

(A,B) → A •s B :=1

2

(A+B)2 − A2 − B2

.

Lemma 1.34(i) The operation •s is symmetric (commutative), i.e., A •s B = B •s A, for all A,B ∈ O.(ii) For all A ∈ O, A •s 0 = 0 •s A = 0.(iii) For all A ∈ O, A •s 1 = 1 •s A = A.

Proof: Clear.

Axiom 11 (Homogeneity)The operation •s is homogenous with respect to the first argument, i.e., (λA)•sB = λ (A •s B),for all A,B ∈ O and λ ∈ R.

Remark 1.35The homogeneity of the operation •s with respect to the second argument is also satisfied becauseof Lemma 1.34, i.e., A •s (λB) = λ (A •s B), for all A,B ∈ O and λ ∈ R.

Lemma 1.36Assume Axioms 2–11. Then, for all A,B ∈ O :(i) A •s B = 1

4

(A+B)2 − (A− B)2

.

(ii) A2 +B2 = 12

(A+B)2 + (A− B)2

.

Proof: (i) Using Axiom 11, note that

2A •s B = A •s B − A •s (−B)

=1

2

(A+B)2 − A2 − B2

−1

2

(A− B)2 − A2 − B2

=1

2

(A+B)2 − (A− B)2

.

12

(ii) Furthermore,

0 = A •s B + A •s (−B)

=1

2

(A+B)2 − A2 − B2

+1

2

(A− B)2 − A2 − B2

=1

2

(A+B)2 + (A− B)2

−A2 +B2

.

Theorem 1.37Assume Axioms 2–11. Then, (O,+, •s) is a distributive and commutative algebra, i.e., for allA,B,C ∈ O, A •s B = B •s A and

(A+B) •s C = A •s C +B •s C.

Proof: From Lemma 1.34 (i), A •s B = B •s A and using Definition 1.33

2(A+B) •s C − 2A •s C − 2B •s C= [(A+B + C)2 + A2] + [B2 + C2]

−[(A+B)2 + (A+ C)2]− (B + C)2.

which by Lemma 1.36 (ii) yields

=1

2[(2A+B + C)2 + (B + C)2]

+1

2[(B + C)2 + (B − C)2]

−1

2[(2A+B + C)2 + (B − C)2]

−(B + C)2 = 0

Remark 1.38Formally, A •s B = 1

2 (AB +BA). The problem is that the product AB is not defined here asit is dependent on the considered physical system.

Theorem 1.39 (Continuity of •S and ()2 in O)Assume Axioms 1–11. Then, we have for all A,B ∈ O that:(i) A •s B ≤ AB.(ii) A2 − B2 ≤ maxA2, B2.(iii) For all convergent sequences An → A in (O, · ), A2

n→ A2 in (O, · ).

13

Proof: (i) By the positivity of states and Axiom 11 combined with Theorem 1.37,

0 ≤ ρ((A+ λB)2) = ρ((A+ λB) •s (A+ λB))

= ρ(A2) + λ2ρ(B2) + 2λρ(A •s B)

for all ρ ∈ E, A,B ∈ O and λ ∈ R. In particular,

|ρ(A •s B)| ≤ 1

2(λ−1A2+ λB2)

=1

2(λ−1A2 + λB2)

for all ρ ∈ E and λ > 0 (cf. Theorem 1.20). If A = 0 or B = 0 then A •s B = 0 and (i)is verified. So, assume that A,B = 0. Upon choosing λ := A/B > 0 we obtain that|ρ(A •s B)| ≤ AB for all ρ ∈ E.

(ii) For all ρ ∈ E,ρ(A2 − B2) = ρ(A2)− ρ(B2)

which implies that

|ρ(A2 − B2)| ≤ maxρ(A2), ρ(B2) ≤ maxA2, B2 = maxA2, B2.

(iii) Using the distributivity, the homogeneity, and the symmetry of •s one gets:

(An + A) •s (An − A)

= An •s An − A •s A− An •s A+ A •s An

= A2n− A2.

Then,

A2n− A2 ≤ An − A(An − A) + 2A

≤ An − A((An − A)+ 2A).

Remark 1.40 (Jordan algebras)The symmetric product is historically associated with Jordan algebras, that is, a distributiveand commutative algebra (X ,+, •s) satisfying the Jordan identity

A •s ((A •s A) •s B) = (A •s A) •s (A •s B)

for all A,B ∈ X . Particular examples of Jordan algebras are C∗–algebras.

Reference: G. Emch. “Algebraic Methods in Statistical Mechanics and Quantum Field The-ory”, Chapter 1.

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1.2 C∗–algebra and its set of states as a model for Axioms 1–11

The aim of this paragraph is to show that Axioms 1–11 are satisfied by a “standard” Model(The space of self–adjoint (s.a.) elements of a C∗–algebra).

1.2.1 C∗–algebras

Definition 1.41 (∗–Algebra)The map A → A∗ from an algebra X to X is an involution, when(i) ∀A ∈ X : (A∗)∗ = A.(ii) ∀A,B ∈ X : (AB)∗ = B∗A∗.(iii) ∀A,B ∈ X , ∀α, β ∈ C: (αA+ βB)∗ = αA∗ + βB∗.An algebra X with an involution is a ∗–algebra and A ∈ X is self–adjoint (s.a.) when A = A∗.

Definition 1.42 (C∗–algebra)A C∗–algebra (X , · , ·, ∗) is a Banach algebra with an involution ∗ such that, for all A ∈ X ,A∗A = A2 with A∗A := A∗ · A.

Remark 1.43 (Properties on C∗–algebras)(i) If the identity 1 ∈ X exists then it is unique, self–adjoint, and 1 = 1.(ii) The set

OX := A = A∗ : A ∈ X,of a C∗–algebra, is a real vector space, cf. Theorem 1.16.(iii) If P(x) =

n

k=0 ckxk is a polynomial with real coefficients ck, then, for all A ∈ OX ,

P(A) :=n

k=0

ckAk ∈ OX .

For all polynomials P1,P2 with real coefficients and all A ∈ OX ,

P1 P2(A) = P1(P2(A)),

see Axiom 2.(iv) For all A,B ∈ OX ,

A •s B :=1

2[(A+B)2 − A2 − B2] =

1

2(AB +BA) ∈ OX ,

see Definition 1.33. In particular, for all A,B ∈ OX and λ ∈ R,

(λA) •s B = A •s (λB) = λ(A •s B),

see Axioms 11.(v) For all A,B ∈ OX ,

A •s B =1

2AB +BA ≤ AB.

15

see Theorem 1.39 (i).(vi) For all A ∈ OX ,

A2 = A∗A = A2,see Theorem 1.20.(vii) Using AB ≤ AB,

AB − AB ≤ A− AB,AB − AB ≤ AB − B.

Then the product (A,B) → AB is also continuous.(viii) The equality A∗A = A2 implies A = A∗ and the continuity of the involution:

A∗ − B∗ = A− B.

(iv) A is invertible iff A∗ is invertible, and then (A−1)∗ = (A∗)−1.

Example 1.44 (Bounded operators)Let H be a Hilbert space. The set (B(H), · op, ·, ∗) of bounded linear operators from H toH is a C∗–algebra with identity. “·” is the standard product of linear operators and, for allA ∈ B(H), A∗ ∈ B(H) is its self–adjoint operator. Here

Aop := supϕ∈H,ϕ=1

Aϕ < ∞

is the operator norm. Let X ⊂ B(H) be any closed subspace such that A∗, AB ∈ X for allA,B ∈ X . Then, (X , · op, ·, ∗) is a C∗–subalgebra of B(H).

Example 1.45 (Continuous functions on compacts)Let K be a compact set and C(K) be the complex vector space of continuous functions from Kto C. For all f ∈ C(K), let

f := supx∈K

|f(x)| < ∞.

Then, (C(K), · ) is a Banach space and by defining the involution by

f ∗(x) := f(x)

and the product byf · g(x) := f(x)g(x),

(C(K), · , ·, ∗) is a commutative C∗–algebra with identity.

Example 1.46 (Measurable bounded functions)Let Ω be a set and B ⊂ 2Ω a σ–algebra. Let M(Ω) be the complex vector space of (B–)measurablefunctions from Ω to C. Using a norm, an involution, and a product as defined in Example1.45, (M(Ω), · , ·, ∗) is a commutative C∗–algebra with identity. If Ω is compact and B is aBorel σ–algebra then C(Ω) ⊂ M(Ω) is a C∗–subalgebra of M(Ω).

16

Notation: From now, X is always a C∗–algebra with identity 1.

Definition 1.47 (Resolvent and spectrum)The resolvent set (A) of A is the set of all λ ∈ C such that (λ1−A) is invertible. The spectrumof A is σ(A) := C\(A). For all λ ∈ (A), the operator

R(A,λ) := (λ1− A)−1 ∈ X

is called the resolvent of A at λ.

Definition 1.48 (Spectral radius)The spectral radius is defined by

r(A) := sup|λ| : λ ∈ σ(A) ≥ 0.

Lemma 1.49 (Spectral radius is well–defined)For all A ∈ X ,

r(A) = infn∈N

An1/n = limn→∞

An1/n ≤ A.

Theorem 1.50 (Spectrum in C∗–algebras)(i) If A ∈ X is normal, i.e., AA∗ = A∗A, then the spectral radius of A is maximal, i.e.,r(A) = A.(ii) If A ∈ X is unitary, i.e., AA∗ = A∗A = 1, then,

σ(A) ⊂ λ ∈ C : |λ| = 1.

(iii) If A ∈ X is self–adjoint, i.e., A = A∗, then

σX (A) ⊂ [−A, A], σX (A2) ⊂ [0, A2].

(iv) If P is a polynomial with complex coefficients, then

σ(P(A)) = P(σ(A)) := P(λ) : λ ∈ σ(A).

Remark 1.51By Theorem 1.50 (iii), σ(A) ∈ R for A = A∗ ∈ X and, by Theorem 1.50 (iv), σ(P(A)) =P(σ(A)) for any polynomial P with real coefficients, see Axiom 3.

Theorem 1.52Let Y ⊂ X be a C∗–subalgebra with identity. For all B ∈ Y, σ(B) = σX (B) = σY(B).

17

Remark 1.53Theorem 1.52 says that the spectrum σ(A) of A ∈ X only depends on the smallest C∗–algebraincluding A, that is,

X (A) := f(A) : f ∈ C(σ(A))see Lemma 6.78.

Definition 1.54 (Representation of C∗–algebras)(i) A pair (π,H), where H and π : X → B(H) are a Hilbert space and a ∗–homomorphismrespectively, is called a representation of X .(ii) A representation (π,H) is faithful if π is injective.(iii) Let (π,H) be a representation of X . If

π(X )Ω = H

for some Ω ∈ H then Ω is called a cyclic vector and the triplet (π,H,Ω) is called a cyclicrepresentation of X .(iv) The representation (π,H) is irreducible if X = 0 and X = H are the unique subspace ofH such that

π(X )X ⊂ X,

i.e., 0 and H are the unique invariant subspace of the representation (π,H).(v) Two representations (π1,H1) and (π2,H2) of X are equivalent if there is a unitary mapU : H1 → H2 such that π1 = U∗π2U .(vi) Two cyclic representations (π1,H1,Ω1) and (π2,H2,Ω2) of X are equivalent if there is aunitary map U : H1 → H2 such that π1 = U∗π2U and Ω2 = UΩ1.

Remark 1.55(i) If (π,H) is an irreducible representation then all vectors Ω ∈ H\0 are cyclic.(ii) [Gelfand–Naimark’s theorem] Any C∗–algebra has a faithful representation.(iii) C∗–algebras admit, in general, infinitely many non–equivalent irreducible representations.

Definition 1.56 (Positive Elements)A ∈ X is positive if A = A∗ and σ(A) ⊂ R+

0 (cf. Lemma 1.14). The set of all positive elementsof X is denoted by X+.

Theorem 1.57(i) X+ is a closed set in X with 0 ∈ X+.(ii) X+ is a cone, i.e., for all α ∈ R+

0 and all A1, A2 ∈ X+: αA ∈ X+ and A1 + A2 ∈ X+.

Definition 1.58 (Order relation)Let A,B ∈ X . A ≥ B if A− B ∈ X+.

Lemma 1.59The relation ≥ is a partial order in X , compare with Axiom 6.

18

Proof: The reflexivity of ≥ follows from the fact that 0 ∈ X+ and the transitivity results fromthe cone properties of X+. Let A,B ∈ X with A ≥ B and B ≥ A. Then,

(A− B) ∈ X+, −(A− B) ∈ X+.

Hence, per Definition, (A− B) = (A− B)∗ and

σ(A− B) ⊂ R+0 , σ(−(A− B)) ⊂ R+

0 .

By Theorem 1.50 (iv), it follows that σ(A − B) = 0, i.e., r(A − B) = 0. By Theorem 1.50(i), the latter yields A− B = 0.

1.2.2 States on C∗–algebras

Notation: Recall that X is a C∗–algebra with identity. Here ρ : X → C is always a continuouslinear functional.

Definition 1.60 (States on C∗–algebras)(i) ρ is Hermitian if

ρ(A) = ρ(A∗), ∀A ∈ X .

(ii) ρ is positive if ρ(A) ≥ 0 for all A ∈ X+. Notation: ρ ≥ 0.(iii) ρ is a state if ρ ≥ 0 and ρ(1) = 1. The set of all states of X is denoted by X ∗

+,1.

Lemma 1.61(i) The Hermitian property of ρ implies

ρ = supρ(A) : A = A∗ ∈ A, A ≤ 1.

(ii) ρ ≥ 0 implies that ρ is Hermitian and ρ(B∗AB) ≤ Aρ(B∗B).

Theorem 1.62A linear functional ρ : X → C is positive when ρ = ρ(1). In particular, positive linearfunctionals on a C∗–algebra are continuous and a linear functional ρ : X → C is a stateprovided ρ = ρ(1) = 1.

Example 1.63Let H be a Hilbert space and X ⊂ B(H) be a C∗–algebra with identity. Then:(i) For any ψ ∈ H with ψ = 1, ρ

ψ: X → C defined by

ρψ(A) = ψ, Aψ

is a state.(ii) Let D ∈ L1(H) (Trace–class operators) with D ≥ 0 and TrD = 1. Then the map ρ

D: X →

C defined byρD(A) = Tr (D · A)

is well–defined as L1(H)B(H) ∈ L1(H) and is clearly a state.19

Example 1.64Let K be a compact set and X = C(K). Let a ∈ K and ρ

a: X → C defined by ρ

a(f) = f(a).

Then ρa(1) = 1 and ρ

a(f ∗f) ≥ 0 for all f ∈ X , i.e., ρ

ais a state of X .

Remark 1.65For the state ρ

a, all f ∈ X = C(K), and all polynomials P, one has ρ

a(P(f)) = P(ρ

a(f)). It

means that ρais dispersion–free. Also, all dispersion–free states of X ∗

+,1 belong to the set

E(X ∗+,1) = ρ

a: a ∈ K.

of all pure states of X ∗+,1.

Example 1.66Let K be a compact set and X = C(K). Let µ be a probability measure on K and the mapρµ: X → C be defined by

ρµ(f) =

K

f(x) dµ(x).

Then ρµ(1) = 1 and ρ

µ(f ∗f) ≥ 0 for all f ∈ X , which means that ρ

µis a state of X .

Theorem 1.67 (Riesz–Markov)Let K be a compact metrizable set, X = C(K) and ρ ∈ X ∗

+,1 be a state. Then there is a uniqueprobability measure µ

ρon K such that, for all f ∈ X ,

ρ(f) =

K

f(x) dµρ(x).

Theorem 1.68 (OX uniquely defines states)Let OX ⊂ X be the real space of s.a. elements of X and O∗

X ,+,1 be the set of all states on OX ,i.e., real linear continuous functionals ρ : OX → R satisfying ρ(A) ≥ 0 for all A ∈ OX ∩ X+

and ρ(1) = 1. Then there is a one–to–one map between any state of X and any state of OX ,i.e., for all ρ ∈ X ∗

+,1, there is a state ρ|OX∈ O∗

X ,+,1 and, for all ρ ∈ O∗X ,+,1, there is a unique

state ρ ∈ X ∗+,1 such that ρ = ρ|OX

.

Proof: For any ρ ∈ X ∗+,1, its restriction ρ|OX

∈ O∗X ,+,1 clearly defines a state on OX . Note that

ρ|OXis a real linear functional because ρ is Hermitian, by Lemma 1.61.

For all A ∈ X , let

ReA : =1

2(A+ A∗) ∈ OX ,

ImA : =1

2i(A− A∗) ∈ OX .

Then,ρ(A) = ρ(AR) + iρ(AI) = ρ|OX

(AR) + iρ|OX(AI),

i.e., ρ|OXuniquely defines ρ. Therefore, for all ρ ∈ O∗

X ,+,1, we uniquely define ρ by

ρ(A) = ρ(AR) + iρ(AI),

which is a Hermitian linear continuous functional from X to C such that |OX= ρ. In particular,

(1) = 1. As X+ ⊂ OX , by Theorem 6.87, ρ is positive.

20

Remark 1.69We identify ρ ∈ X ∗

+,1 with ρ|OX. Moreover, from the last theorem, it follows that O := OX and

E := X ∗+,1 satisfy Axioms 9 and 10.

Lemma 1.70(i) Let Y be another C∗–algebra with identity, ρ : Y → C be a state and ϕ : X → Y be a∗–homomorphism. Then ρ : X → C, ρ = ρ ϕ, is a state.(ii) Let Y be a C∗–subalgebra with identity (i.e. 1X ∈ Y) and ρ : Y → C be state. Then thereis ρ : X → C such that ρ|Y = ρ.(iii) Let A ∈ X and a ∈ σ(A). Then there is a state ρ : X → C such that ρ(A) = a.

Proof: (i) ρ is continuous as ρ and ϕ are both continuous (cf. Theorem 6.75 (i)). Moreover,

ρ(1X ) = ρ(ϕ(1X )) = ρ(1Y) = 1,

i.e., ρ is normalized. For all A ∈ X ,

ρ(A∗A) = ρ(ϕ(A∗A)) = ρ(ϕ(A)∗ϕ(A)) ≥ 0,

i.e., ρ is positive.

(ii) As ρ is a state, by Theorem 1.62, we have ρ = ρ(1) = 1. By Hahn–Banach’s theorem,there is a linear functional ρ : X → C with ρ = ρ(1) = 1, which in turn implies that ρ is astate, by Theorem 1.62.

(iii) Let XA := lin1, A ∈ X be the linear subspace generated by 1, A. We define thelinear functional ρ : XA → C by

ρ(α1+ βA) := α + βa, α, β ∈ C, a ∈ σ(A).

Since α + βa ∈ σ(α1+ βA) for all α, β ∈ C and a ∈ σ(A), we have

|ρ(α1+ βA)| = |α + βa| ≤ r(α1+ βA) = α1+ βA.

This means that ρ ≤ 1. As ρ(1) = 1, ρ = 1. By Hahn–Banach’s theorem, there is a linearfunctional ρ : X → C with ρ = ρ(1) = 1 and ρ(A) = a. By Theorem 1.62, ρ is a state.

Theorem 1.71 (States separates points)(i) Equalities ρ(A) = 0 for all ρ ∈ X ∗

+,1 imply that A = 0.(ii) ρ(A) ∈ R for all ρ ∈ X ∗

+,1 yields A∗ = A.(iii) ρ(A) ≥ 0 for all ρ ∈ X ∗

+,1 yields A ∈ X+.(iv) If A ∈ X is normal (i.e., AA∗ = A∗A) then there is ρ ∈ X ∗

+,1 such that |ρ(A)| = A.

Proof: Use Lemma 1.70.

21

Remark 1.72Let O := OX and E := X ∗

+,1.(i) Theorem 1.71 (i) implies that :(a) The addition in Axiom 7 must correspond to the addition in X . In particular, Axiom 7holds.(b) The multiplication by a scalar in O (see Definition 1.2) must be the one of X .(c) The constant observable λ ∈ O for λ ∈ R (see Lemma 1.3) equals λ = λ1, with 1 being theidentity of X .(ii) Theorem 1.71 (iii) implies that the order relation of Definition 1.10 corresponds to the orderrelation in X (Definition 1.58). In particular, by Lemma 1.59, Axiom 6 holds.(iii) It follows that positive (cf. Definition 1.13) elements of O are the elements X+ ⊂ OX = O.(iv) Theorem 1.71 (iv) implies that the norm · in O of Definition 1.17 is the C∗–Norm ofX .

1.2.3 GNS (Gelfand-Neumark-Segal) representation of states

We show here that the Hilbert space structure can be derived from the so–called GNS repre-sentation of states defined as follows: First, observe that a positive linear functional satisfiesthe Cauchy–Schwarz inequality:

Lemma 1.73 (Cauchy–Schwarz)Let X be a C∗–algebra with identity and ρ be a positive linear functional X → C. Then, for allA,B ∈ X ,

|ρ(B∗A)|2 ≤ ρ(A∗A)ρ(B∗B).

Proof: For any A ∈ X , A∗A ∈ X+ and ρ(A∗A) ≥ 0, by positivity of ρ ≥ 0. Since ρ is Hermitian,it follows that the equation

A,Bρ = ρ(B∗A), A,B ∈ X ,

defines an inner product on X . Therefore, this lemma corresponds to the Cauchy–Schwarzinequality

|A,Bρ|2 ≤ A,AρB,Bρ, A,B ∈ X .

Therefore, if X is a C∗–algebra with identity and ρ ∈ X ∗+,1 then

Lρ := A ∈ X : ρ(A∗A) = 0

is a closed left–ideal of X , i.e., Lρ is a closed subspace such that XLρ ⊂ Lρ, cf. Lemma 1.61.The equation

A+ Lρ, B + Lρρ = ρ(B∗A)

defines the scalar product ·, ·ρ in X /Lρ. For all A ∈ X ,

πρ(A)(B + Lρ) = AB + Lρ

22

defines a linear operator πρ(A) : X /Lρ → X/Lρ with

πρ(A)X/Lρ := supB∈X/Lρ

ρ (B∗A∗AB)

ρ (B∗B)≤ AX ,

by Lemma 1.61.

In particular, let

Hρ : = X /Lρ

·,·ρ

Ωρ : = (1+ Lρ) ∈ Hρ

and define the mapπρ : X → B(Hρ)

via the continuous extension ofπρ(A)|X/Lρ = πρ(A).

Then, (πρ,Hρ,Ωρ) is the GNS–representation of ρ:

Theorem 1.74 (GNS–representation)Let X be a C∗–algebra with identity and let ρ ∈ X ∗

+,1. There is a cyclic representation(πρ,Hρ,Ωρ) such that

ρ(A) = Ωρ, πρ(A)Ωρfor all A ∈ X . Up to a unitary transformation this representation is unique.

Example 1.75Let H be a Hilbert space and X = B(H), cf. Example 1.63. Then:(i) For any ψ ∈ H with ψ = 1, let ρ

ψ: X → C be the state defined by ρ

ψ(A) = ψ, Aψ. Then

A,Bψ := ρψ(B∗A) = Bψ, Aψ, A,B ∈ X

andLψ := A ∈ X : Aψ = 0.

In particular,

Hψ := X /Lψ

·,·ψ ∼ H, Ωψ ∼ ψ ∈ H,

and the map πρ : X → B(H) is defined by πψ(A) = A. (“∼” formally means that there is aunitary transformation).(ii) Let D ∈ L1(H) (Trace–class operators) with D ≥ 0, TrD = 1 and let ρ

D: X → C be the

state defined by

ρD(A) = Tr (D · A) =

N

n=1

pnψn, Aψ

n

with N > 0, pn > 0 and ψnNn=1 ⊂ H being linearly independent normalized vectors (ψ

n = 1).

Then

A,BD := Tr (D · B∗A) =N

n=1

pnBψn, Aψ

n, A,B ∈ X

23

andLD :=

n∈1,...,NLψn

In particular,

HD := X /LD

·,·D ∼N

⊕n=1

H, ΩD ∼N

⊕n=1

ψn

and the map πD : X → B(N

⊕n=1

H) is defined in by

πD(A)

N

⊕n=1

ϕn

=

N

⊕n=1

(Aϕn) .

Remark 1.76The state ρ ∈ X ∗

+,1 is faithful if Lρ = 0. This implies that the GNS–representation is alsofaithful. Moreover, if a state ρ is faithful then X /Lρ = X and A,Bρ = ρ(B∗A) for allA,B ∈ X ⊂ Hρ. Equilibrium states in quantum statistical mechanics are usually faithful.

1.2.4 Commutative C∗–algebras and dispersion–free states

Definition 1.77 (Spectrum of a C∗–algebra)Let X be a commutative C∗–algebra. A linear functional ρ : X → C is a character if for allA,B ∈ X ,

ρ(AB) = ρ(A)ρ(B).

The set σ(X ) of all characters of X is called the spectrum of X .

Lemma 1.78Let X be a commutative C∗–algebra with identity. Then σ(X ) ⊂ X ∗

+,1 and, for all ρ ∈ σ(X )and A ∈ X , ρ(A) ∈ σ(A).

Proof: Let ρ ∈ σ(X ). Then, for all A ∈ X ,

ρ(A) = ρ(A1) = ρ(A)ρ(1).

As ρ = 0, ρ(1) = 0 and since,ρ(1n) = ρ(1)n

for n ∈ N, we have ρ(1) = 1. Take A ∈ X and λ /∈ σ(A). Then, for some B ∈ X ,

(λ1− A)B = 1

and so,(λ− ρ(A))ρ(B) = 1.

In particular, ρ(A) = λ for all λ /∈ σ(A), i.e., ρ(A) ∈ σ(A). Since ρ(A) ∈ σ(A) for all A ∈ X+,ρ(A) ≥ 0, i.e., ρ is positive. So, ρ is a state.

24

Remark 1.79Let X = C(K) and, for a ∈ K, take the state ρ

a∈ X ∗

+,1 of Example 1.64. ρais clearly a

character. For all f ∈ X , σ(f) = f(K). For all λ ∈ σ(f), let a ∈ f−1(λ). Then, ρa(f) = λ,

i.e., for all f ∈ X and λ ∈ σ(f), there is ρ ∈ σ(X ) such that ρ(f) = λ.

Theorem 1.80Let X be a commutative C∗–algebra with identity and A = A∗ ∈ X . Then, for all a ∈ σ(A),there is ρ ∈ σ(X ) such that ρ(A) = a.

Remark 1.81Let X be a C∗–algebra with identity and A = A∗ ∈ X . Let

X (A) := f(A) : f ∈ C(σ(A))

be the smallest commutative C∗–algebra including A, see Remark 1.53. Then:(i) By Lemma 1.70 (ii), all characters ρ ∈ σ(X (A)) has an extension ρ ∈ X ∗

+,1, which isdispersion–free w.r.t. A, i.e., ρ(P(A)) = P(ρ(A)) for all polynomials P. By Theorem 1.80, wehave that, for all a ∈ σ(A), there is a dispersion–free (w.r.t. A) ρ ∈ X ∗

+,1 such that ρ(A) = a.(ii) Conversely, for all dispersion–free (w.r.t. A) ρ ∈ X ∗

+,1, ρ|X (A) ∈ σ(X (A)). In particular, byLemma 1.78, ρ(A) ∈ σ(A) for all dispersion–free (w.r.t. A) ρ ∈ X ∗

+,1.

1.2.5 Conclusion:

Let X be a C∗–algebra with identity and define that:

(i) O := OX , E = O∗X ,+,1 := X ∗

+,1.

(ii) For all A ∈ O ⊂ X , let WA := σ(A). By Theorem 1.50 (iii), WA ⊂ R for all A ∈ O.

(iii) For all polynomials P(x) =

n

k=0 ckxk with real coefficients and for all A ∈ O ⊂ X ,

P(A) :=n

k=0

ckAk ∈ O ⊂ X .

(iv) For all A,B ∈ O, let A+B ∈ O be defined by the addition in X .

Then, from (i)–(iii), Axioms 1–11 holds:

Axiom 1: By Theorem 1.50 (iii), for all A ∈ O,

WA = σ(A) ⊂ [−A, A]

and A < ∞.

Axiom 2: This follows from the definition (iii) and Remark 1.43 (ii).

25

Axiom 3: This results from Theorem 1.50 (iv).

Axiom 4: (i) By Theorem 1.50 (iii) and the linearity of states it suffices to prove that, for all A ∈ Owith infWA = 0,

0 ≤ ρ(A) ≤ supWA

for all ρ ∈ E. If infWA = 0 then A ∈ X+ and since states are positive functionals,ρ(A) ≥ 0 for all ρ ∈ E. If A ∈ X+ then

A = sup σ(A) = supWA.

Since ρ = 1, by Theorem 1.62 for ρ ∈ X ∗+,1, ρ(A) ≤ supWA for all ρ ∈ E and A ∈ X+.

(ii) This results from linearity.

Axiom 5: By Remark 1.81, σE(A) = σX (A) for all A ∈ O.

Axiom 6: By Remark 1.72 the order relation ≥ in O (Definition 1.10) is the order relation ≥ of X(Definition 1.58). By Lemma 1.59 it is a partial order.

Axiom 7: This results from the linearity of states.

Axiom 8: This results from the completeness of X and the continuity of the involution in X .

Axiom 9: This results from the fact that states of E are functionals on O.

Axiom 10: See Remark 1.69.

Axiom 11: See Remark 1.43 (iv).

Remark 1.82The set of all s.a. elements of a C∗–algebra is not the unique structure from which one canobtain Axioms 1–11. There are, for instance, Jordan algebras which cannot be represented bys.a. elements of a C∗–algebra. However, we only consider here the C∗–algebraic case.

1.3 Classical systems versus quantum ones

1.3.1 Classical systems

Definition 1.83 (Classical systems)A physical system S is said to be a classical system if O = OX with X being a commutativeC∗–algebra.

Theorem 1.84 (Gelfand–Naimark)Let X be a commutative C∗–algebra with identity. Then the set σ(X ) ⊂ X ∗

+,1 of all charactersof X is weak∗–compact and there is a ∗–isomorphism from C(σ(X )) to X .

26

Remark 1.85(i) Theorem 1.84 means that – up to ∗–isomorphisms – every commutative C∗–algebra X hasthe form X = C(K) with K being a compact set, i.e., it is the commutative C∗–algebra ofExample 1.45.(ii) By the Riesz–Markov theorem (Theorem 1.67), we can identify every state ρ of O = OXwith a probability measure µ

ρon K.

(iii) By Remark 1.79, the set

EO :=

A∈OX

EA

of all dispersion–free states (with respect to) w.r.t. to any element of O = OX are the statesof Example 1.64. In particular, all points of K can be identified with a dispersion–free stateρ ∈ EO. The states of EO are the pure states of X ∗

+,1, see Remark 1.65.

Definition 1.86 (Configuration space)Let S be a classical system with O = OX and X = C(K). The set K is called the configurationspace (or phase space) of the physical system S.

1.3.2 Quantum systems and Heisenberg uncertainty principle

Definition 1.87 (Quantum systems)A physical system S is said to be a quantum one if

EO :=

A∈OEA = ∅.

Definition 1.88 (Fluctuation of observables)Let S be a physical system, A ∈ O be an observable, and ρ ∈ E be a state of S. Then,

∆ρ(A) :=ρ((A− ρ(A))2)

is called the fluctuations of A w.r.t. ρ.

Lemma 1.89 (Heisenberg uncertainty principle)Let S be a physical system with O = OX , E = X ∗

+,1, where X is a C∗–algebra. Then, for allstates ρ ∈ E and A,B ∈ O,

∆ρ(A)∆ρ(B) ≥ 1

2|ρ([A,B])|,

where[A,B] := AB − BA ∈ X .

Proof: Assume without loss of generality (w.l.o.g.) that ρ(A) = ρ(B) = 0. By Lemma 1.73(Cauchy–Schwarz),

|ρ([A,B])| ≤ |ρ(AB)|+ |ρ(BA)| ≤ 2ρ(A2)ρ(B2).

27

Remark 1.90(i) Let S be a physical system with O = OX , E = X ∗

+,1, where X is a C∗–algebra. If there istwo observables A,B ∈ O such that

ρ([A,B]) = 0,

for all ρ ∈ E, then, by Lemma 1.89, S is a quantum system.(ii) For all A,B ∈ O = OX , [A,B] ∈ X but [A,B] /∈ O.

Example 1.91Let X := B(C2) and define

A :=

0 11 0

, B :=

0 i−i 0

, C :=

1 00 −1

.

Then,

[A,B] = −2i

1 00 −1

,

i.e., [A,B] = −2iC. In particular, ρ(C) = 0 for all ρ ∈ EO. Now, let ρ ∈ X ∗+,1 such that

ρ(C) = 0. Sinceρ(C2) = ρ(1) = 1 = 0

ρ /∈ EO, i.e., EO = ∅.

Remark 1.92It follows that S is a quantum system with O = OX when the C∗–algebra X includes the matrixalgebra B(C2) as a C∗–subalgebra.

1.4 C∗–dynamical systems

1.4.1 Heisenberg picture of quantum mechanics (QM)

Definition 1.93 (Autonomous C∗–dynam. syst.) Let X be C∗–algebra with identity 1. Afamily αtt∈R of automorphisms of X is called a one–parameter group of automorphisms when:(i) α0 = 1.(ii) For all t, s ∈ R, αt αs = αs+t.In this case, the pair (X , αtt∈R) is called an autonomous C∗–dynamical system when, for allA ∈ X , αt(A) defines a continuous function from R to X .

Example 1.94Let K be a compact set and X = C(K) (classical system). Let ξ

tt∈R be a family of homo-

morphisms from K to K such that(i) ξ0(p) = p for all p ∈ K.(ii) For all t, s ∈ R, ξ

t ξ

s= ξ

t+s.

(iii) The map R×K → K, (t, p) → ξt(p) is continuous.

28

Define, for all t ∈ R, the map αt : X → X by

αt(f)(p) := f ξt(p).

Then, (X , αtt∈R) is an autonomous C∗–dynamical system [Proof: use the compacticity of K].

Definition 1.95 (Non–autonomous C∗–dynam. syst.) Let X be C∗–algebra with identity1. A family αtt∈R of automorphisms of X is called a two–parameters group of automorphismswhen:(i) For all t ∈ R+

0 , αt,t = 1.(ii) For all t, r, s ∈ R+

0 , t ≥ r ≥ s, αt,s = αt,r αr,s.In this case, the pair (X , αt,st,s∈R+

0 , t≥s) is called a non–autonomous C∗–dynamical system

when, for all A ∈ X , (t, s) → αt,s(A) is a continuous map from

∆ := (t, s) ∈ R+0 × R+

0 : t ≥ s

to X .

Definition 1.96 (Generator)Let (X , αtt∈R) be an autonomous C∗–dynamical system. Define the linear subspace

D(G) := A ∈ X : t → αt(A) is diff. at t = 0 ⊂ X

and the linear operator G : D(G) → X by

G(A) :=d

dt

t=0

αt(A)

The operator G is called the generator of the C∗–dynamical system (X , αtt∈R) and D(G) isthe domain of definition of G.

Remark 1.97(i) The generator G of a C∗–dynamical system (X , αtt∈R) is unique and will be identified witha (microscopic) interaction. D(G) is generally a dense ∗–subalgebra of X .(ii) For all A0 ∈ D(G) and t > 0, A(t) := αt(A0) ∈ D(G) and

d

dtA(t) = G(A(t)).

Definition 1.98 (Fundamental solution)A non–autonomous C∗–dynamical system (X , αt,st,s∈R+

0 , t≥s) is called a fundamental solution

of the initial value problem

A(t) = G(t)(A(t)) , t > s,A(t) = A0 , t = s,

when, for all A0 ∈ X and s ≥ 0, the function t → A(t) := αt,s(A0) is differentiable for all t > sand

d

dtA(t) = G(t)(A(t))

29

for all t > s. Here G(t) : D(G(t)) → X is a linear operator for which

A(t) = αt,s(A0) ∈ D(G(t)),

for all A0 ∈ D(G(s)), and s ∈ R+0 with t > s.

Remark 1.99The linear operator G(t) : D(G(t)) → X is (uniquely) defined and will be identified with a(microscopic) interaction.

Lemma 1.100Let H1 and H2 be two Hilbert spaces and X1 ⊂ B(H1), X2 ⊂ B(H2) two C∗–algebras withidentity. Let α(1)

t t∈R and α(2)t t∈R be two family of automorphisms such that (X1, α(1)

t t∈R)and (X2, α(2)

t t∈R) are autonomous C∗–dynamical systems. Define the C∗–algebra X ⊂ B(H1⊗H2) by

X := linA1 ⊗ A2 : A1 ∈ X1, A2 ∈ X2·B(H1⊗H2) .

If X1 or X2 is finite dimensional then there is a unique family αtt∈R of automorphisms of Xsuch that

αt(1X1 ⊗ A2) = 1X1 ⊗ α(2)t (A2),

αt(A1 ⊗ 1X2) = α(1)t (A1)⊗ 1X2 ,

for all t ∈ R, A1 ∈ X1 and A2 ∈ X2. Moreover, (X , αtt∈R) defines a C∗–dynamical system.

Remark 1.101A similar result for two non–autonomous C∗–dynamical systems also holds.

1.4.2 Schrodinger picture of quantum mechanics

Definition 1.102Let (X , αtt∈R) be an autonomous C∗–dynamical system and a state ρ ∈ X ∗

+,1. For all t ∈ R,let the state

ρt:= ρ αt ∈ X ∗

+,1.

[cf. Lemma 1.70 (i)]. Similarly, for a non–autonomous C∗–dynamical system

(X , αt,st,s∈R+0, t ≥ s)

and ρ ∈ X ∗+,1, define for all t ≥ s the state

ρt:= ρ αt,0 ∈ X ∗

+,1.

Then ρtis called the state of (X , αtt∈R) (resp. (X , αt,st,s∈R+

0, t ≥ s)) at time t. The state

ρ is stationary if ρt= ρ for all times t.

30

Remark 1.103Let (X , αtt∈R) be an autonomous C∗–dynamical system and ρ ∈ X ∗

+,1. Per definition, for allt ∈ R and A ∈ X ,

ρ(αt(A)) = ρt(A).

The same holds for non–autonomous ones.

Lemma 1.104Every C∗–dynamical system (X , αtt∈R) has at least one stationary state ρ ∈ X ∗

+,1.

Idea of the proof: Define, for all n ∈ N, the state ρn∈ X ∗

+,1 by

ρn(A) :=

1

n

n

0

ω αt(A)dt, A ∈ X .

Observe that this integral is well–defined, by continuity of the map t → αt(A), as a Riemannintegral. As the set of all states is weak∗–compact, there is a sequence ρ

nk(or a net) which is

convergent in the weak∗–topology to some

ρ := limk→∞

ρnk.

which turns out to be a stationary state of (X , αt).

Theorem 1.105Let (X , αtt∈R) be an autonomous C∗–dynamical system and ρ ∈ X ∗

+,1 be a stationary statewith GNS–representation (Hρ, πρ,Ωρ). Then, there is a unique family U(t)t∈R of unitaryoperators Hρ → Hρ such that:(i) Ωρ = U(t)Ωρ for all t ∈ R.(ii) For all A ∈ X and t ∈ R,

πρ(αt(A)) = U(t)πρ(A)U(t)∗.

This family is strongly continuous, i.e., for all x ∈ Hρ, the map t → U(t)x is a continuousfunction from R to Hρ.

Proof: Define the subspaceH := πρ(X )Ωρ ⊂ Hρ.

By cyclicity of Ωρ, H is a dense set in Hρ. For all t ∈ R, we also define the linear operator

U(t) : H → H

byU(t)πρ(A)Ωρ = πρ(αt(A))Ωρ

for all A ∈ X . Becauseαt(α−t(A)) = 1,

31

we haveRan(U(t)) = H

is a dense set. Using that

πρ(αt(A))Ωρ, πρ(αt(A))Ωρ= Ωρ, πρ(αt(A))

∗πρ(αt(A))Ωρ= Ωρ, πρ(αt(A

∗)(αt(A))Ωρ= Ωρ, πρ(αt(A

∗A))Ωρ= Ωρ, πρ(αt(A

∗A))Ωρ= ρ(αt(A

∗A)) = ρ(A∗A)

= πρ(A)Ωρ, πρ(A)Ωρ,

the operator U(t) is an isometry. In particular, there is a unique unitary extension U(t) : Hρ →Hρ of U(t). Moreover, (i) results from

U(t)Ωρ = U(t)πρ(1)Ωρ = πρ(αt(1))Ωρ = πρ(1)Ωρ = Ωρ.

Then, by definition of Ut and using that U(t)∗ = U(−t),

U(t)πρ(A)U(t)∗πρ(αt(B))Ωρ

= U(t)πρ(A)U(t)∗U(t)πρ(B)Ωρ,

= U(t)πρ(A)πρ(B)Ωρ,

= U(t)πρ(AB)Ωρ,

= πρ(αt(AB))Ωρ

= πρ(αt(A))πρ(αt(B))Ωρ.

In other words, for all x ∈ H,

U(t)πρ(A)U(t)∗x = πρ(αt(A))πρ(αt(B))x.

Upon choosing B = α−t(C) and x = πρ(C)Ωρ ∈ H, it follows that

U(t)πρ(A)U(t)∗x = πρ(αt(A))x.

As H is dense we then obtain

U(t)πρ(A)U(t)∗ = πρ(αt(A))

in the sense of operators.

Let V (t)t∈R be a family of unitary operator on Hρ which satisfies (i) and (ii). Then, forall A ∈ X ,

V (t)πρ(A)Ωρ = V (t)πρ(A)V (t)∗V (t)Ωρ = πρ(αt(A))Ωρ,

i.e., U(t) and V (t) are equal to each others on a dense set H. This implies that U(t) = V (t).

32

It remains to show that, for all x ∈ Hρ, t → U(t)x is a continuous function. Since U(t) = 1for all t, it suffices to show this on the dense set H.

U(t)πρ(A)Ωρ − U(t)πρ(A)Ωρ2

= (U(t)− U(t))πρ(A)Ωρ, (U(t)− U(t))πρ(A)Ωρ= πρ(αt(A)− αt(A))Ωρ, πρ(αt(A)− αt(A))Ωρ.

Then, since t → αt(A) is continuous,

limt→t

U(t)πρ(A)− U(t)πρ(A)Ωρ = 0,

Corollary 1.106Let (X , αtt∈R) be an autonomous C∗–dynamical system and ρ ∈ X ∗

+,1 be a stationary state withGNS–representation (Hρ, πρ,Ωρ). Then there is a unique (generally unbounded) self–adjoint op-erator Hρ, called the Hamiltonian of (X , αtt∈R) in the vacuum representation of the stationarystate ρ, such that, for all t1, . . . , tn ∈ R, n ∈ N, and A1, . . . , An ∈ X ,

G(A1,t1),...,(An,tn) : = ρ(αt1(A1) · · ·αtn(An))

= Ωρ, πρ(A1)ei(t1−t2)Hρπρ(A2) · · ·

· · · ei(tn−1−tn)Hρπρ(An)Ωρ.

Proof: See FA-III.

Proof: The strongly continuous unitary family U(t)t∈R of Theorem 1.105 is a unitary group:as in the proof of Theorem 1.105, using

U(t+ t)πρ(A)Ωρ = πρ(αt+t(A))Ωρ

= πρ(αt(αt(A)))Ωρ

= U(t)πρ(αt(A))Ωρ

= U(t)U(t)πρ(A)Ωρ

for all A ∈ X and t, t ∈ R we obtain the equality U(t + t) = U(t)U(t) for t, t ∈ R. Thenthere is self–adjoint operator H : Hρ ⊃ D(H) → Hρ such that U(t) = e−itH for all t ∈ R. SeeFA-III.

Remark 1.107The cyclic representation (Hρ, πρ,Ωρ) as well as the Hamiltonian Hρ, can generally directly befound from the Green functions G(A1,t1),...,(An,tn) of the C∗–dynamical system. This is the aim of“constructive quantum mechanics” (or more generally, “constructive quantum field theory”).

33

2 Equilibrium State of Finite Systems

2.1 Finite systems and Gibbs state

Definition 2.1 (Physical system)A physical system S is said to be finite when O = OX , with X being a C∗–algebra isomorph toa matrix algebra B(Cn) for some n ∈ N.

Definition 2.2 (Hamiltonian of finite systems)Let S be a finite system with O = OX . The observable H ∈ OX which represents the totalenergy of the system S is called the Hamiltonian of S.

Lemma 2.3 (Density matrix of states)Let S be a finite system with O = OX , X = B(Cn), n ∈ N. Let D ≥ 0 be a positive element ofX with Tr D = 1. Then

ρD(A) = Tr(D · A), A ∈ X

defines a state, see Example 1.63. Conversely, for all states ρ ∈ E, there is a unique positiveelement Dρ ∈ X , the so–called density matrix of ρ, such that Tr D = 1 and ρ = ρ

Dρ.

Proof: DefineA,B := Tr(A∗B)

for all A,B ∈ X . Then H = (X , ·, ·) is a Hilbert space. The state ρ defines a continuouslinear functional from H to C. Via Frechet–Riesz’s representation, there is a unique Dρ ∈ Xsuch that

ρ(A) = Dρ, A = Tr(D∗ρA),

for all A ∈ X . In particular,

Tr(D∗ρ) = Tr(D∗

ρ1) = ρ(1) = 1.

Since ρ is a positive functional,

Tr(PD∗ρP ) = Tr(D∗

ρP ) = ρ(P ) ≥ 0,

for all orthogonal projections P ∈ B(Cn). In particular,

x,D∗ρxCn ≥ 0

for all x ∈ Cn, i.e., D∗ρis a positive matrix. In particular, D∗

ρ= Dρ.

Definition 2.4 (Gibbs state)Let S be a finite system with O = OX , X = B(Cn), n ∈ N and Hamiltonian H ∈ OX . TheGibbs state at inverse temperature β ∈ (0,∞) is the unique state ωH,β defined by the densitymatrix

DH,β := Z−1H,β

exp(−βH),

whereZH,β := Tr (exp(−βH)) .

34

Remark 2.5The Gibbs state ωH,β represent in physics the thermodynamic equilibrium of the system attemperature T = β−1.

2.2 Gibbs state as minimizer of the free–energy – Maximum entropyprinciple

Definition 2.6 (von Neumann entropy on B(Cn))Let S be a finite system with O = OX , X = B(Cn), n ∈ N. We define the continuous functionη : [0,∞) → R by

η(x) := −x log(x), x > 0, η(0) := 0.

For any state ρ ∈ E, its von Neumann entropy is

SB(Cn)(ρ) = S(ρ) := Tr η(Dρ) ∈ [0, log(n)].

Remark 2.7Let ρ be a state on B(Cn) with density matrix Dρ. Let Pρ = p1, . . . , pn be the eigenvalue ofDρ. As Dρ ≥ 0 and Tr Dρ = 1, pi ≥ 0 and

pi = 1. So, Pρ can be seen as probability measure

on 1, 2, . . . , n and

S(ρ) = −n

i=1, pi>0

pi log(pi)

is the Shannon entropy of Pρ. In particular, S(ρ) is a measure of disorders of the state ρ.

Definition 2.8 (von Neumann entropy)Let X be a C∗–Algebra ∗–isomorph to B(Cn) via the map ϕ : A → B(Cn). Then, for allρ ∈ X ∗

+,1, the von Neumann entropy is defined by

SX (ρ) := SB(Cn)(ρ ϕ−1).

Lemma 2.9 (Well–definedness of the von Neumann entropy)Let X be a C∗–Algebra ∗–isomorph to B(Cn) and ϕ,ϕ : X → B(Cn) be two ∗–isomorphisms.Then any state ρ ∈ X ∗

+,1 defines two states ρ, ρ on B(Cn) by

ρ := ρ ϕ−1, ρ := ρ ϕ−1

and there is a unitary matrix U ∈ B(Cn) such that Dρ = U∗DρU . In particular, SX (ρ) doesnot depend on the ∗–isomorphism.

Idea of the proof: For any normalized vector Ω ∈ Cn (Ω = 1), there is a normalized vectorΩ ∈ Cn (Ω = 1) such that, for all A ∈ X ,

Ω,ϕ(A)Ω = Ω,ϕ(A)Ω =: ρ(A).

35

Since any non–zero vector Φ = 0 ∈ Cn is cyclic w.r.t. ϕ(X ) and w.r.t. ϕ(X ), (ϕ,Cn,Ω) and(ϕ,Cn,Ω) are two GNS–representations of the state ρ ∈ X ∗

+,1. By Theorem 1.74, they areunitarily equivalent, i.e., there is a unitary matrix U such that

ϕ(A) = Uϕ(A)U∗

for all A ∈ X . Therefore, for any A ∈ B(Cn),

Tr((U∗DρU)A) = Tr((Dρ(UAU∗))

= Tr((Dρ(Uϕ((ϕ−1(A))U∗))

= Tr((Dρϕ((ϕ−1(A)))

= ρ ϕ−1 ϕ((ϕ−1(A))

= ρ(A)

= Tr(DρA).

Note that U∗DρU ≥ 0 and Tr(U∗DρU) = 1 and, by uniqueness of the density matrix, we haveU∗DρU = Dρ . In particular,

Tr (η(Dρ)) = Tr (η(UDρU∗)) = Tr (Uη(Dρ)U

∗) = Tr (U∗Uη(Dρ)) = Tr (η(Dρ)) .

Definition 2.10 (Free energy)Let S be a finite system with O = OX , X = B(Cn), n ∈ N and Hamiltonian H ∈ OX . The freeenergy Fβ(ρ) of any state ρ ∈ X ∗

+,1 at inverse temperature β ∈ (0,∞) is defined by

Fβ(ρ) := ρ(H)− β−1S(ρ).

Theorem 2.11 (Passivity of Gibbs states)For any β ∈ (0,∞) and H ∈ OX

infρ∈X ∗

+,1

Fβ(ρ) = Fβ(ωH,β) = −β−1 logZH,β.

Proof: Let ejnj=1 ⊂ Cn be an orthonormal basis (ONB) of eigenvector of Dρ with corre-sponding eigenvalue pjnj=1: Dρ(ej) = pjej. Without loss of generality, assume that pj > 0.Then

−βFβ(ρ) = −βρ(H) + S(ρ)

= −β

j

pjej, Hej −

j

pj log(pj)

=

j

pj[−βej, Hej − log(pj)].

If ρ = ωH,β then ejnj=1 ⊂ Cn is an ONB of eigenvectors of H and Dρ with correspondingeigenvalue hjnj=1 for H. Therefore,

pj = Z−1H,β

e−βhj .36

In particular,

−βFβ(ωH,β) =

j

pj[−βhj − (−βhj − logZH,β)]

= logZH,β.

Using Jensen’s inequality, for any ρ ∈ X ∗+,1,

exp(−βFβ(ρ)) ≤

j

pj exp[−βej, Hej − log(pj)]

=

j

pjp−1j

exp[−βej, Hej]

≤

j

ej, exp(−βH)ej

= exp(logZH,β),

Using Peierls–Bogoliubov’s inequality, that is,

n

k=1

φ(ek, Aek) ≤n

k=1

ek,φ(A)ek = Tr φ(A)

for A = A∗ ∈ B(Cn), any convex function φ : R → R and with eknk=1 ⊂ Cn be an orthonormalbasis, we deduce that

Fβ(ρ) ≥ −β−1 logZH,β.

Remark 2.12 (Maximum entropy principle)Let S be a finite system with Hamiltonian H ∈ OX . Let

Eβ := ωH,β(H)

(Energy of the system in the Gibbs state). Then

infρ∈X ∗

+,1

βFβ(ρ) = infρ∈X ∗

+,1

(βρ(H)− S(ρ))

= infρ∈X ∗

+,1, ρ(H)=Eβ

(βρ(H)− S(ρ))

= βEβ − supρ∈X ∗

+,1, ρ(H)=Eβ

S(ρ)

= βEβ − S(ωH,β).

This means that ωH,β maximizes the entropy S over ρ ∈ X ∗+,1 : ρ(H) = Eβ, i.e., under the

constraint that the energy is fixed. The corresponding Lagrange multiplier is β.

37

2.3 Gibbs states as tangent functionals

Definition 2.13 (Pressure)Let S be a finite system with Hamiltonian H ∈ OX . The pressure is defined by

PH,β := β−1 logZH,β

at inverse temperature β > 0.

Theorem 2.14 (Convexity and continuity of the pressure)Let X = B(Cn). Then the map H → PH,β from OX to R has the following properties:(i) It is convex, i.e., for all H1, H2 ∈ OX and all λ ∈ [0, 1],

PλH1+(1−λ)H2,β ≤ λPH1,β + (1− λ)PH2,β.

(ii) It is Lipschitz continuous: For all H1, H2 ∈ OX ,

|PH1,β − PH2,β| ≤ H1 −H2.

Proof: (i) For all ρ ∈ X ∗+,1, define the affine functional aρ : OX → R by

aρ(A) := −ρ(A) + β−1S(ρ).

By Theorem 2.11,

PλH1+(1−λ)H2,β = supρ∈X ∗

+,1

aρ(λH1 + (1− λ)H2)

= supρ∈X ∗

+,1

(λaρ(H1) + (1− λ)aρ(H2))

≤ λ supρ∈X ∗

+,1

aρ(H1) + (1− λ) supρ∈X ∗

+,1

aρ(H2)

= λPH1,β + (1− λ)PH2,β.

(ii) By Theorem 2.11 (i.e., Bogoliubov’s inequality),

|PH1,β − PH2,β| ≤ max |ωH1,β(H2 −H1)|, |ωH2,β(H2 −H1)|≤ H1 −H2.

Definition 2.15 (Tangent functionals)Let X be a real vector space and g : X → R a convex function. The linear functional ξ : X → Ris tangent to g at x ∈ X if, for all y ∈ X ,

g(x+ y)− g(x) ≥ ξ(y).

Lemma 2.16Let S be a finite system with Hamiltonian H ∈ OX . For any minimizer ω ∈ X ∗

+,1 of the freeenergy FH,β, −ω is tangent to the map H → PH,β at H ∈ X .

38

Proof: As ω is a minimizer of FH,β, we have from Theorem 2.11 that

PH+H,β − PH,β = − infρ∈X ∗

+,1

FH+H,β(ρ) + FH,β(ω)

≥ −FH+H,β(ω) + FH,β(ω)

= −ω(H )− FH,β(ω) + FH,β(ω).

Corollary 2.17 (Gibbs states as tangent functionals)Let S be a finite system with Hamiltonian H ∈ OX . Then −ωH,β is tangent to the mapH → PH,β at H ∈ X .

Lemma 2.18Let S be a finite system with OS = OX . For all H1, H2 ∈ OX ,

d

dt

s=0

PH1+sH2,β = −ωH1,β(H2).

Proof: Letf(s) = Tr (exp(−β(H1 + sH2))) .

Then,PH1+sH2,β = −β−1 log f(s).

Moreover, because of the cyclicity of the trace (Tr(·)),

f(s) = Tr

∞

n=0

1

n!(H1 + sH2)

n(−β)n

=∞

n=0

1

n!Tr ((H1 + sH2)

n(−β)n)

= f(0)− β∞

n=1

Tr

n1

n!sHn−1

1 (−β)n−1H2

+O(s2)

= f(0)− βsTr(exp(−βH1)H2) +O(s2).

Therefore,d

dt

s=0

f(s) = −β ZH1,β ωH1,β(H2).

Corollary 2.19Let S be a finite system with OS = OX and the pressure P·,β as the map OX → R, H → PH,β.Then, P·,β is differentiable on OX and

dPH,β = −ωH,β.

In particular, −ωH,β is the unique tangent functional at H to the pressure H → PH,β and so,the Gibbs state ωH,β is the unique minimizer of the free energy FH,β (cf. Lemma 2.16).

39

Remark 2.20Gibbs state can also be seen as a KMS (Kubo-Martin-Schwinger) state, see Theorem 6.111. Itis also related to the zero–law of thermodynamics, see Theorem 6.116. For more details, seeSection 6.6.

Remark 2.21The conditions:(i) Minimizer of the free energy,(ii) Tangent to the pressure,on a state has been written without using a representation of the C∗–algebra. This is notalways possible as the Gibbs state cannot always be defined, for instance when the spectrum ofthe Hamiltonian H is continuous for which e−βH is not trace class.

3 Infinite Lattice Systems

3.1 Spin and fermion algebras on lattices

3.1.1 Spin algebras

Definition 3.1 (Lattice)For all d ∈ N, L = Ld := Zd is the quadratic lattice of dimension d. Pf = Pf (L) ⊂ 2L is theset of all finite subsets of L. For all ∈ N, let Λ ∈ Pf be the box

Λ := (x1, . . . , xd) ∈ L : |xi| ≤ .

Definition 3.2 (Local spin algebras)Let the Hilbert space H = CS with S being a finite set (of spins). For any x ∈ L, let US

x be(disjoint) copies of the C∗–algebra B(H). For all Λ ∈ Pf ,

US

Λ :=

x∈LUS

x

is the local spin algebra of the (finite) set Λ ⊂ L.

Remark 3.3(i) Let Λ,Λ ∈ Pf with Λ ⊂ Λ. Then

US

Λ ≡ US

Λ ⊗ US

Λ\Λ.

In particular, A ∈ UΛ can be identified with A⊗ 1USΛ\Λ

and US

Λ ⊂ US

Λ for all Λ ⊂ Λ ∈ Pf .

(ii) Let Λ,Λ ∈ Pf be disjoint sets. Then, for all A ∈ US

Λ ⊂ US

Λ∪Λ and all B ∈ US

Λ ⊂ US

Λ∪Λ,

[A,B] := AB − BA = 0.

40

Definition 3.4 (Global spin algebra)The spin algebra of the lattice L is the closure

US := US

0

·

of the ∗–subalgebraUS

0 :=

Λ∈Pf

US

Λ

of local elements.

Lemma 3.5 (Translation automorphisms)For all A ∈ B(CS) and all x ∈ L, let Ax ∈ US

x be the copy of A ∈ US

x := B(CS). For all

x ∈ Zd, the conditionαx(Ay) = Ax+y, A ∈ B(CS), y ∈ L, (3.1)

uniquely defined a ∗–automorphism of US, called the translation automorphism of x ∈ Zd.

Idea of the proof: Condition (3.1) shows that αx must be fixed on a dense set US

0 . As ∗–automorphisms are continuous, it follows that only one ∗–automorphism can satisfy (3.1). Wethen construct, for any x ∈ Zd, a sequence of unitary operators Ux, ∈ US

Λsuch that

U∗x,AyUx, = Ax+y, y ∈ L,

for all sufficiently large . This implies the existence of a bijective ∗–homomorphism αx : US

0 →US

0 satisfying (3.1) with αx = 1. The unique extension αx of αx on US satisfies (3.1) and isan isometric ∗–homomorphism with αx(US) = US.

3.1.2 Fermion algebras

Lemma 3.6Let S be a finite set (of spins). For all Λ ∈ Pf , there are a C∗–algebra UF

Λ := B(C2Λ×S) and

cx,s ∈ UF

Λ , x ∈ Λ, s ∈ S such that

X (cx,s : x ∈ Λ, s ∈ S) = UF

Λ

andc∗

x,s, cx,s = δx,xδs,s1,

cx,s, cx,s = 0.(3.2)

Here A,B := AB + BA. The last relation is called the anticommutation relations (CAR)and implements the Pauli principle for fermions. cx,s ∈ B(C2L×S

) are uniquely defined up to aunitary operator. For all Λ ⊂ Λ ∈ Pf , there is an injective ∗–homomorphism jΛ,Λ : UF

Λ → UF

Λ

such that jΛ,Λ(cx,s) = cx,s, for all x ∈ Λ and s ∈ S.

41

Remark 3.7(i) From the CAR,

c∗x,scx,s + cx,sc

∗x,s

= 1.

As c∗x,scx,s and cx,sc∗x,s are positive elements, one gets that

c∗x,scx,s, cx,sc∗x,s ≤ 1.

In fact, we havec∗

x,scx,s, cx,sc∗x,s = 1.

(ii) Because of the existence of the isometric ∗–homomorphism jΛ,Λ, UF

Λ can be seen as asubalgebra of UF

Λ for Λ ⊂ Λ ∈ Pf .(iii) Let Λ ⊂ Λ ∈ Pf . The algebras UF

Λ and UF

Λ ⊗ UF

Λ\Λ are ∗–isomorphic, but there is no∗–isomorphism

jΛ,Λ : UF

Λ ⊗ UF

Λ\Λ → UF

Λ

such thatjΛ,Λ(UF

Λ ⊗ 1UFΛ\Λ

) ⊂ UF

Λ ⊂ UF

Λ

andjΛ,Λ(1UF

Λ⊗ UF

Λ\Λ) ⊂ UF

Λ\Λ ⊂ UF

Λ .

(iv) Let Λ,Λ ∈ Pf be two disjoint sets and A ∈ UF

Λ ⊂ UF

Λ∪Λ, B ∈ UF

Λ ⊂ UF

Λ∪Λ. Then, ingeneral, [A,B] = 0.

Definition 3.8 (Global fermion algebras)The fermion algebra of the lattice L is the closure

UF := UF

0

·

of the ∗–subalgebraUF

0 :=

Λ∈Pf

UF

Λ

of local elements.

Observe that, by Lemma 3.6, jΛ,Λ(cx,s) = cx,s for all Λ ⊂ Λ ∈ Pf , x ∈ Λ and s ∈ S. And

jΛ,Λ jΛ,Λ = jΛ,Λ

for all Λ ⊂ Λ ⊂ Λ ∈ Pf .

Lemma 3.9 (Separability of US/F )The C∗–algebras US and UF are both separable.

Proof: Observe thatUS/F

0 =

∈NUS/F

Λ.

It means that US/F

0 is a countable union of finite dimensional spaces. In particular , US/F

0 hasa dense countable set D. As US/F

0 is dense in US/F , D is also dense in US/F .

42

Lemma 3.10 (Translation and gauge automorphism)(i) For all x ∈ Zd, the condition

αx(cy,s) = cx+y,s, s ∈ S, y ∈ L,

uniquely defines a ∗–automorphism of UF .(ii) For all ϕ ∈ [0, 2π), the conditions

αϕ(cx,s) = eiϕcx,s, s ∈ S, x ∈ L

uniquely defines a ∗–automorphism of UF , called a (global) gauge automorphism of the fermionfield algebra.

Proof: As in Lemma 3.5.

Definition 3.11 (Even subalgebra)The set

UF,+ := A ∈ UF : απ(A) = +A.is the C∗–subalgebra of even elements. The set

UF,− := A ∈ UF : απ(A) = −A

is the space of odd elements.

3.2 The set E1 of translation invariant (t.i.) states

Definition 3.12 (t.i. states)The set of all translation invariant (t.i.) states on US/F is defined by

E1 :=

x∈Zd, A∈US/F

ρ ∈ (US/F )∗ : ρ(1) = 1, ρ(A∗A) ≥ 0 with ρ = ρ αx.

Remark 3.13(i) Using Corollary 6.30 E1 ⊂ (US/F )∗+,1 is convex, weak–∗ compact, and metrizable. So, byKrein–Milman’s theorem (Theorem 1.31), it is the weak∗–closure of the convex hull of the (non–empty) set E (E1) of its extreme points.(ii) A t.i. state ρ ∈ E1 on UF is automatically even, i.e., ρ απ = ρ.(iii) Let Λ,Λ ∈ Pf be two disjoint sets and ρ ∈ (UF

Λ )∗+,1, ρ

∈ (UF

Λ)∗+,1 be two even states. Thenthere is a unique state ρ⊗ ρ ∈ (UF

Λ∪Λ)∗+,1 such that

ρ⊗ ρ(AB) = ρ(A)ρ(B)

for all A ∈ UF

Λ ⊂ UF

Λ∪Λ and B ∈ UF

Λ ⊂ UF

Λ∪Λ. Observe that ρ⊗ ρ is a tensor product of statesin the standard sense (see Remark 3.7 (iii)).

43

For all ∈ N and A ∈ US/F , let

A :=1

|Λ|

x∈Λ

αx(A).

Definition 3.14 (Ergodic states)A state ρ ∈ E1 is ergodic if

lim→∞

ρ(A∗A) = |ρ(A)|2

for all A ∈ US/F .

Theorem 3.15 (von Neumann ergodic theorem)For any t.i. state ρ ∈ E1,

lim→∞

ρ(A∗A) ∈ [|ρ(A)|2, A2].

Proof: We restrict, without loss of generality, to the one–dimensional case L = Zd=1.

(a) Let (πρ,Hρ,Ωρ) be the GNS representation of ρ. Since ρ is t.i., i.e., ρ αx = ρ for allx ∈ Zd=1, there is a unitary family Uxx∈Zd , Ux : Hρ → Hρ, such that (i) UxΩρ = Ωρ forall x ∈ Zd; (ii) πρ(αx(A)) = Uxπρ(A)U∗

xfor all x ∈ Zd and all A ∈ US/F ; (iii) U0 = 1,

UxUy = Ux+y for all x, y ∈ Zd. This is proven as in the proof of Theorem 1.105.

(b) Observe that, as in Theorem 6.80 in the case of self–adjoint operators, for any normaloperator A acting on a Hilbert space, there is a unique measurable functional calculus.

Since U1 is unitary, U1 is normal and σ(U1) ⊂ K1 := z ∈ C : |z| = 1. See Theorem1.50 (ii).

(c) For all A ∈ US/F and ∈ N,

|Λ|2ρ(A∗A)

=

Ωρ,

x∈Λ

πρ(αx(A∗))

y∈Λ

πρ(αy(A))Ωρ

=

x∈Λ

πρ(αx(A))Ωρ,

y∈Λ

πρ(αy(A))Ωρ

=

x∈Λ

πρ(αx(A))Ωρ

2

=

x∈Λ

Uxπρ(A)U∗xΩρ

2

=

x∈Λ

Uxπρ(A)Ωρ

2

= |Λ|2 Pπρ(A)Ωρ2 ,44

where

P :=1

|Λ|

x∈Λ

Ux.

Let the continuous function p : K1 ⊃ σ(U1) → C be defined by

p(z) :=1

|Λ|

x∈Λ

zx.

for all ∈ N. Using the functional calculus of U1,

ρ(A∗A) = p(U1)πρ(A)Ωρ2 .

(d) Note that p ≤ 1 for all ∈ N and p → p∞ pointwise, where p∞(1) := 1 and p∞(z) := 0for z ∈ σ(U1)\1. Observe that 1 ∈ σ(U1) because U1Ωρ = Ωρ. Using the functionalcalculus,

lim→∞

p(U1)πρ(A)Ωρ = p∞(U1)πρ(A)Ωρ.

In particular, the desired limit exists:

lim→∞

ρ(A∗A) = p∞(U1)πρ(A)Ωρ2.

Since p∞ = 1,p∞(U1)πρ(A)Ωρ2 ≤ πρ(A)Ωρ2 ≤ A2.

(e) Because Ran(p∞) ⊂ 0, 1, one has p∞(U1)p∞(U1) = p∞(U1) and p∞(U1)∗ = p∞(U1). Inother words, p∞(U1) is a orthogonal projection. Moreover, as UxΩρ = Ωρ for all x ∈ Zd,by using the functional calculus, p∞(U1)Ωρ = Ωρ. Then

|ρ(A)|2 = ρ(A)ρ(A) = ρ(A∗)ρ(A)

= Ωρ, πρ(A∗)ΩρΩρ, πρ(A)Ωρ

= Ωρ, πρ(A∗)p∞(U1)Ωρp∞(U1)Ωρ, πρ(A)Ωρ

= Ωρ, πρ(A∗)p∞(U1)ΩρΩρ, p∞(U1)πρ(A)Ωρ

= Ωρ, πρ(A∗)p∞(U1)PρPρp∞(U1)πρ(A)Ωρ

= Pρp∞(U1)πρ(A)Ωρ, Pρp∞(U1)πρ(A)Ωρ= Pρp∞(U1)πρ(A)Ωρ2.

Here Pρ is the orthogonal projection with Ran (Pρ) = CΩρ. In particular,

|ρ(A)|2 ≤ p∞(U1)πρ(A)Ωρ2.

Remark 3.16If ρ ∈ E1 is ergodic then, per definition,

lim→∞

ρ(A2) = ρ(A)2 = ρ(A)

2

for all A = A∗ ∈ US/F . In other words, ρ is dispersion–free w.r.t. A in the limit → ∞ andthus, there is a “(quantum) law of large numbers” for ρ.

45

Lemma 3.17Any ergodic state ρ is extremal in E1.

Proof: If ρ /∈ E(E1) is not extreme, there are two states ρ1, ρ2 ∈ E1 with ρ = 12ρ1 +

12ρ2 and

ρ1(A) = ρ2(A) for some A = A∗ ∈ US/F . By convexity of the real function x → |x|2 ,

|ρ(A)|2 =

1

2ρ1(A) +

1

2ρ2(A)

<1

2|ρ1(A)|2 +

1

2|ρ2(A)|2

≤ 1

2lim→∞

(ρ1(A2) + ρ2(A

2))

= lim→∞

ρ(A2).

Corollary 3.18 (Density of the set E (E1))The set E (E1) is a Gδ weak∗–dense subset of E1.

Proof: The set E (E1) of extreme points of E1 is a Gδ set, by Theorem 6.35 (i), as E1 ismetrizable. Thus, it suffices to prove that E(E1) is dense in E1. For any ρ ∈ E1, we define thestate ρ

nto be the restriction ρΛn

∈ EΛn on the box

Λn :=x = (x1, · · · , xd) ∈ Zd : |xi| ≤ n

(3.3)

seen as a (2n+ 1)(1, ..., 1)–periodic state, i.e.,

ρn:=

x∈(2+1)Zd

ρ|UΛn+x ,

This is possible, by [6, Theorem 11.2.], because any t.i. state is even, by Remark 3.13 (ii).From the state ρ

n∈ E(2n+1)(1,...1) we define next the t.i. state

ρn:=

1

|Λn|

x∈Λn

ρn αx ∈ E1. (3.4)

Clearly, the space–averaged state ρnconverges towards ρ ∈ E1 w.r.t. the weak∗–topology and

we prove below that ρn∈ E(E1) by using Lemma 3.17.

Indeed, for any A ∈ U0, there is a positive constant C > 0 such that

ρn(αx(A

∗)αy(A)) = ρn(αx(A

∗)) ρn(αy(A))

whenever d(x, y) ≥ C. Here, d : L × L → [0,∞) is the usual metric defined on the latticeL := Zd. Using the space–average A we then deduce that

ρn(A∗

A) =

1

|Λ|2

x,y∈Λ

ρn(αx(A

∗)) ρn(αy(A)) (3.5)

+O(−d).46

Since ρn∈ E1 is a t.i. state, for any A ∈ U0, one has that

1

|Λ|

x∈Λ

ρn(αx(A)) = ρ

n(A) +O(−1)

which combined with the asymptotics (3.5) implies that

lim→∞

ρn(A∗

A) = |ρ

n(A)|2 .

Using this last equality we then obtain from (3.4) that, for any A ∈ U0,

limL→∞

ρn(A∗

A) = |ρ

n(A)|2 (3.6)

because ρn∈ E1 and

αx(A∗A) = (αx (A))

∗(αx (A)) (3.7)

for all x ∈ Zd. Since the set U0 is dense in the fermion algebra U , we can extend (3.6) to anyA ∈ U which shows that the state ρ

n∈ E1 is ergodic and thus extreme in E1, by Lemma 3.17.

Theorem 3.19 (Extremality = Ergodicity)Any extreme state ρ ∈ E(E1) of E1 is ergodic and conversely.

Pure states are thus the states satisfying the law of large numbers, i.e., the fluctuations ofthe measure vanishes in the limit by space–averaging.

Theorem 3.20 (Ergodic decomposition of states in E1)For any ρ ∈ E1, there is a unique probability measure µ

ρon E1 such that

µρ(E1) = 1 and ρ =

E1

dµρ(ρ) ρ.

Furthermore, the map ρ → µρis an isometry in the norm of linear functionals, i.e., ρ− ρ =

µρ− µ

ρ for any ρ, ρ ∈ E1.

Proof: See Choquet’s theorem (Theorem 6.40). Uniqueness of the probability measure µρuses

Theorem 3.19 and von Neumann ergodic theorem, see Theorem IV.4.1 in R.B. Israel, Convexityin the theory of lattice gases, Princeton Univ. Press, 1979.

Corollary 3.21The set E1 is affinely homeomorphic to the Poulsen simplex.

Proof: This follows from Corollary 3.18 and Theorem 3.20 combined with Theorem 6.48.

Remark 3.22An example of a closed face of E1 is given by the set EΠ ⊂ E1 of permutation invariant states.This set is a Bauer simplex, which is, in a sense, complementary to the Poulsen simplex.Extreme states of EΠ are product states and conversely, i.e., E(EΠ) = E⊗. This is Størmer the-orem which is a non–commutative version of the celebrated de Finetti Theorem from (classical)probability theory.

47

3.3 Interactions

Definition 3.23 (Interaction)Let U = US/F . The map Φ : Pf → U is an interaction if the following conditions hold:

(i) For all Λ ∈ Pf , Φ(Λ) = Φ(Λ)∗ ∈ US/F

Λ .(ii) If U = UF , Φ(Λ) ∈ UF,+ for all Λ ∈ Pf .It is finite range if:(iii) There is R < ∞ such that Φ(Λ) = 0 for all Λ ∈ Pf with

d(Λ) = maxx− y : x, y ∈ Λ > R.

It is translation invariant (t.i.) when:(iv) For all Λ ∈ Pf and x ∈ Zd:

αx(Φ(Λ)) = Φ(Λ+ x),

whereΛ+ x := y + x : y ∈ Λ.

Definition 3.24 (Banach space of t.i. interactions)The real Banach space W1 is the set of all t.i. interactions Φ with finite norm

ΦW1 :=

Λ∈Pf , 0∈Λ|Λ|−1Φ(Λ) < ∞.

Remark 3.25(W1, · ) is a Banach space and the subspace

W f1 := Φ ∈ W1 : Φ finite range

is dense in W1.

3.4 Energy density

Definition 3.26(i) For Φ ∈ W1 and all Λ ∈ Pf ,

UΦΛ :=

Λ∈Pf , Λ⊂Λ

Φ(Λ) ∈ US/F

Λ ⊂ US/F .

is called the energy observable of the finite set Λ ⊂ L.(ii) The observable

uΦ :=

Λ∈Pf , 0∈Λ|Λ|−1Φ(Λ) ∈ OU

is called the energy density observable.3.28

48

Remark 3.27uΦ is well–defined because uΦUS/F ≤ ΦW1 < ∞.

Lemma 3.28For Φ ∈ W1, ∈ N and all ρ ∈ E1,

lim→∞

1

|Λ|ρ(U) = ρ(uΦ).

with U := UΦΛ

∈ US/F

Λ.

Proof: For all Φ ∈ W1,

UΦ

=

Λ∈Pf

1Λ⊂ΛΦ(Λ)

=

x∈Λ∩Zd

Λ∈Pf ,x∈Λ

1

|Λ|1Λ⊂ΛΦ(Λ)

=

x∈Λ∩Zd

Λ∈Pf ,0∈Λ

1Λ⊂Λ−xΦ(x+ Λ)

|Λ| .

Then, for all t.i. states ρ ∈ E1,

ρUΦΛ

|Λ|=

Λ∈Pf ,0∈Λ

ρ

Φ(Λ)

|Λ|

x∈Λ∩Zd

1Λ⊂Λ−x

|Λ|

and ρΦ(Λ)

|Λ|

x∈Λ∩Zd

1Λ⊂Λ−x

|Λ|

≤ Φ(Λ)|Λ|

for any Λ ∈ Pf and all . Moreover, for any Λ ∈ Pf ,

lim→∞

x∈Λ∩Zd

1Λ⊂Λ−x

|Λ|

= 1.

As

Λ∈Pf , 0∈Λ

Φ(Λ)|Λ| = ΦW1 < ∞

we can use Lebesgue’s dominated convergence theorem to obtain that

lim→∞

1

|Λ|ρ(U) =

Λ∈Pf , 0∈Λ

1

|Λ|ρ(Φ(Λ))

= ρ

Λ∈Pf , 0∈Λ

1

|Λ|Φ(Λ)

= ρ(uΦ).

49

Definition 3.29 (Energy density functional)The energy density functional eΦ : E1 → R w.r.t. a t.i. local interaction Φ ∈ W1 is defined by

eΦ(ρ) := ρ(uΦ) < ∞.

Remark 3.30(i) eΦ : E1 → R is affine: For all ρ, ρ ∈ E1 and all λ ∈ [0, 1],

eΦ(λρ+ (1− λ)ρ) = λeΦ(ρ) + (1− λ)eΦ(ρ).

(ii) eΦ : E1 → R is weak–∗ continuous. See Corollary 6.26 (ii).(iii) For all Φ,Ψ ∈ W1 and ρ ∈ E1,

|eΦ(ρ)− eΨ(ρ)| ≤ Φ−ΨW1 .

3.5 Entropy density

The von Neumann entropyS(ρΛ) := Tr

η(DρΛ

)≥ 0. (3.8)

(η(x) := −x log(x)) has the following well–known properties (cf. Section 6.5.5):

S1 It is strongly sub–additive, i.e., for any Λ1,Λ2 ∈ Pf (L) and any local state ρΛ1∪Λ2on

UΛ1∪Λ2 ,S(ρΛ1∪Λ2

)− S(ρΛ1)− S(ρΛ2

) + S(ρΛ1∩Λ2) ≤ 0.

S2 It is concave, i.e., for any Λ ∈ Pf (L), any states ρΛ,1, ρΛ,2 on UΛ, and λ ∈ [0, 1],

S(λρΛ,1 + (1− λ)ρΛ,2) ≥ λS(ρΛ,1) + (1− λ)S(ρΛ,2).

S3 It is approximately convex, i.e., for any Λ ∈ Pf (L), any states ρΛ,1, ρΛ,2 on UΛ, andλ ∈ [0, 1],

S(λρΛ,1 + (1− λ)ρΛ,2) ≤ λS(ρΛ,1) + (1− λ)S(ρΛ,2) + η(λ) + η(1− λ).

S1–S3 ensure the existence of the entropy density s : E1 → R+0 :

Theorem 3.31 (Existence of the entropy density)For any t.i. state ρ ∈ E1,

s(ρ) := lim→∞

1

|Λ|S(ρ|UΛ

) = inf∈N

1

|Λ|S(ρ|UΛ

)

.

defines an affine map s from E1 to R, called the entropy density functional.

50

Proof: Let

s : = lim inf→∞

1

|Λ|S(ρ|UΛ

),

s : = lim sup→∞

1

|Λ|S(ρ|UΛ

).

We need to prove that

s = s = inf

1

|Λ|S(ρ|UΛ

) : ∈ N.

Since

s ≥ s ≥ inf

1

|Λ|S(ρ|UΛ

) : ∈ N,

it suffices to prove that

s ≤ inf

1

|Λ|S(ρ|UΛ

) : ∈ N.

For any > 0, let ∈ N such thatinf

1

|Λ|S(ρ|UΛ

) : ∈ N− 1

|Λ |S(ρ|UΛ

)

< /2.

Without loss of generality, let L = Zd=1. Because of the sub–additivity S1 of the Von–Neumann–entropy and the translation invariance of ρ

1

|Λ|S(ρ|UΛ

) ≤ 1

|Λ|

Λ

Λ

S(ρ|UΛ

) + const

.

Here [x] ∈ N0 for x ∈ R+0 is the largest integer such that [x] ≤ x. In particular,

1

|Λ|S(ρ|UΛ

) ≤ 1

|Λ |S(ρ|UΛ

) +

2

for sufficiently large . It follows, for all > 0,

s ≤ inf

1

|Λ|S(ρ|UΛ

) : ∈ N+ .

The concavity of s follows from the concavity S2 of the Von–Neumann–entropy. Since, forall λ ∈ (0, 1),

lim→∞

η(λ) + η(1− λ)

|Λ|= 0,

the convexity of s results from the quasi–convexity S3 of the von–Neumann–entropy.

Remark 3.32 (Entropy density for periodically inv. states)Let L = (L1, . . . , Ld) ∈ Nd. A state ρ ∈

US/F

∗+,1

is L–periodic if ρ αx = ρ for all x ∈ L.Zd.

Let ELbe the set of L–periodic states. In the same way one proves Theorem 3.31, one gets, for

all ρ ∈ EL,

s(ρ) := lim→∞

1

|Λ|S(ρ|US/F

Λ

) = inf∈L.Nd

1

|Λ|S(ρ|US/F

Λ

)

.

The entropy density functional ρ → s(ρ) on ELstays affine. Moreover, it is translation invari-

ant on EL: s(ρ) = s(ρ αx) for all x ∈ Zd (ρ αx ∈ EL

).51

Theorem 3.33The entropy density functional s : E1 → R is upper–semi–continuous (u.s.c.) in the weak∗–topology.

Proof: Define, for all ∈ N, the function s : E1 → R by

s(ρ) :=1

|Λ|S(ρ|US/F

Λ

).

The functional s is, for all ∈ N, weak∗–continuous. (use Theorem 6.25 to show that E1 →(US/F

Λ)∗, ρ → ρ|US/F

Λ

is weak∗–continuous.) By Theorem 3.31

−s(ρ) = − infs(ρ) : ∈ N= sup−s(ρ) : ∈ N

Therefore, for all c ∈ R,(−s)−1((c,∞)) =

∈N(−s)

−1((c,∞)).

In particular, for all c ∈ R, (−s)−1((c,∞)) is an open set and (−s) is lower–semi–continuous.

4 Equilibrium States of Infinite Systems

4.1 Equilibrium states as minimizers of the free–energy density

Definition 4.1 (Free–energy density functional fΦ)For β ∈ (0,∞], the free–energy density functional fΦ w.r.t. the t.i. interaction Φ ∈ W1 is themap

ρ → fΦ(ρ) := eΦ(ρ)− β−1s(ρ)

from E1 to R.

Definition 4.2 (Set of t.i. equilibrium states)For β ∈ (0,∞) and any Φ ∈ W1, the set MΦ of t.i. equilibrium states is the set

MΦ := ω ∈ E1 : fΦ (ω) = inf fΦ(E1)

of all minimizers of the free–energy density functional fΦ over the set E1.

Definition 4.3 (Faces)A subset S of a convex set K is a face of K if S = ∅, S is convex, and the assertion

x = λx1 + (1− λ)x2 ∈ S, λ ∈ (0, 1), x1, x2 ∈ K

implies x1, x2 ∈ S.52

It is equivalent to say that all extreme points of S are also extremal in K.

Lemma 4.4MΦ is weak∗–closed face of E1. In particular, MΦ is compact.

Proof: Because E1 is metrizable, it is sequentially compact and we can restrict ourself onsequences instead of more general nets.

MΦ = ∅: Take any sequence ρn∞n=1 of approximating t.i. minimizers, that is, any sequence

ρn∞n=1 in E1 such that

limn→∞

fΦ(ρn) = inf fΦ(E1).

Since E1 is compact and metrizable, we can assume without loss of generality that ρnconverges

to ω ∈ E1 as n → ∞. Since eΦ is continuous and −s is u.s.c. in the weak∗–topology,

fΦ(ω) ≤ limn→∞

fΦ(ρn) = inf fΦ(E1),

i.e., ω ∈ MΦ. Thus, MΦ = ∅.Compacticity of MΦ: Since E1 is compact, MΦ ⊂ E1 is compact if it is a closed set which is

proven as above.

Convexity of MΦ: Let ω,ω ∈ MΦ. Since the functional fΦ is affine,

fΦ(λω + (1− λ)ω) = λfΦ(ω) + (1− λ)fΦ(ω) = inf fΦ(E1)

for all λ ∈ [0, 1].

MΦ is a face: Let ω ∈ MΦ such that ω = λω1 + (1 − λ)ω2 with λ ∈ (0, 1) and ω1,ω2 ∈ E1.By affinity of fΦ,

fΦ(ω) = inf fΦ(E1) = λfΦ(ω1) + (1− λ)fΦ(ω2),

which implies ω1,ω2 ∈ MΦ.

Remark 4.5Since MΦ is a face of E1, extreme states of MΦ are ergodic, see Theorem 3.19.

4.2 Gibbs states versus equilibrium states

Let

p = p(Φ) :=1

β|Λ|log Tr(e−βU

ΦΛ )

= −|Λ|−1Fβ(ωUΦΛ

,β) = − infρ∈

US/FΛ

∗

+,1

|Λ|−1Fβ(ρ)

be the finite volume pressure associated with the internal energy UΦΛ

for β ∈ (0,∞) and anyΦ ∈ W1, see Theorem 2.11. In the limit → ∞ it defines a map Φ → P (Φ) from W1 to R:

53

Definition 4.6 (Pressure)For β ∈ (0,∞), the (infinite volume) pressure is the map from W1 to R defined by

Φ → P (Φ) := lim→∞

p(Φ).

Theorem 4.7 (Pressure as a variational problem on states)(i) For any Φ ∈ W1,

P (Φ) = − inf fΦ(E1) < ∞.

(ii) The map Φ → P (Φ) from W1 to R is Lipschitz continuous: For all Φ,Φ ∈ W1,

|P (Φ)− P (Φ)| ≤ Φ− Φ.

Proof:

(a) SinceUΦ

Λ− UΦ

Λ ≤ |Λ|Φ− Φ,

by Theorem 2.14 (ii), for all ∈ N,

|p(Φ)− p(Φ)| ≤ Φ− Φ.

In particular,|P (Φ)− P (Φ)| ≤ Φ− Φ.

By Definition of eΦ(ρ) one gets, for all ρ ∈ E1,

|eΦ(ρ)− eΦ(ρ)| = |fΦ(ρ)− fΦ(ρ)| ≤ Φ− Φ.

which implies that| inf fΦ(E1)− inf fΦ(E1)| ≤ Φ− Φ.

Therefore, it suffices to prove the theorem for Φ in the dense subset W f1 of finite range

interactions.

(b) For any ρ ∈ E1, by Theorem 2.11,

−p =1

|Λ|FU

ΦΛ

,β(ωUΦΛ

,β) ≤1

|Λ|FU

ΦΛ

,β(ρ|US/FΛ

)

which yieldslim sup→∞

(−p) ≤ fΦ(ρ)

for all ρ ∈ E1. Therefore,lim sup→∞

(−p) ≤ inf fΦ(E1).

54

(c) For all ∈ N, let ω be the unique (2+ 1)(1, 1, . . . , 1)–periodic product state

ω =

x∈(2+1)Zd

ω|US/FΛ+x

,

such thatω|US/F

Λ

= ωUΦΛ

,β.

By additivity of the Von–Neumann–entropy w.r.t. product states,

s(ω) =1

|Λ|S(ωU

ΦΛ

,β).

Let ω ∈ E1, ∈ N, be defined by

ω :=1

|Λ|

x∈Λ

ω αx.

Since the entropy density is affine and translation invariant (see Remark 3.32), one has

s(ω) =1

|Λ|

x∈Λ

s(ω αx) =1

|Λ|S(ωU

ΦΛ

,β).

(d) For any finite range interaction Φ ∈ W f1,

1

|Λ|UΦΛ

=1

|Λ|

x∈Λ

αx(uΦ) +O(−1).

Then

eΦ(ω) = ω(uΦ)

=1

|Λ|

x∈Λ

ω(αx(uΦ))

=1

|Λ|ω(U

ΦΛ) +O(−1)

=1

|Λ|ωU

ΦΛ

,β(UΦΛ) +O(−1).

It follows that

fΦ(ω) =1

|Λ|FU

ΦΛ

,β(ωUΦΛ

,β) +O(−1) = −p +O(−1)

which in turn implies that

lim inf→∞

(−p) = lim inf→∞

(fΦ(ω)). (4.9)

(e) From (b) and (d) we obtain− lim

→∞p = inf fΦ(E1). (4.10)

55

Theorem 4.8 (Weak∗–limit of space–averaged t.i. Gibbs states)For any Φ ∈ W1, the weak∗–accumulation points of the sequence ωl∈N of ergodic states

ω :=1

|Λ|

x∈Λ

x∈(2+1)Zd

ωUΦΛ+x,β

αx ∈ E(E1)

belong to the set MΦ of equilibrium states.

Proof: Since E1 is compact and metrizable, the sequence ωl∈N has (possibly few) weak∗–accumulation points ω and by l.s.c. of fΦ together with (4.9)–(4.10), one gets ω ∈ MΦ. Notethat ω ∈ E(E1), see proof of Corollary 3.18.

There are, however, important differences between the finite volume system and its thermo-dynamic limit:

• Non–uniqueness of equilibrium states. The Gibbs state is the unique minimizer of thefinite volume free–energy density (Theorem 2.11) but at infinite volume, ω may not beunique:

Lemma 4.9 (Ergodic states as t.i. equilibrium states)For any finite subset ρ1, . . . , ρn ⊂ E1 of ergodic states, there is Φ ∈ W1 such thatρ1, . . . , ρn ⊂ MΦ.

Mathematically, it is related to the fact that we leave the Fock space representation of modelsto go to a representation–free formulation of thermodynamic phases, see Section 1.1. Doingso we take advantage of the non–uniqueness of the representation of the C∗–algebra US/F .This property is, indeed, necessary to get non–unique equilibrium states which imply phasetransitions.

• Space symmetry of equilibrium states. The Gibbs state minimizes the finite volume free–

energy density over the setUS/F

Λ

∗

+,1of all states (Theorem 2.11). But it may not

converge to a t.i. state in the thermodynamic limit.

4.3 Equilibrium states as tangent functionals

4.3.1 The set E1 as a subset of the dual space W∗1

Definition 4.10We define the map ξ from the set E1 of t.i. states to the dual space W∗

1 by

ξ(ρ)(Φ) := −eΦ(ρ),

for all ρ ∈ E1 and Φ ∈ W1.

56

Theorem 4.11(i) ξ is affine, i.e.,

ξ(λρ1 + (1− λ)ρ2) = λξ(ρ1) + (1− λ)ξ(ρ2)

for all ρ1, ρ2 ∈ E1 and λ ∈ [0, 1].(ii) ξ : (E1, · ) → (W∗

1 , · ) is an isometry, i.e., for all ρ, ρ ∈ E1,

ξ(ρ)− ξ(ρ)W∗1= ρ− ρ(US/F )∗ .

(iii) ξ : (E1, τ ∗|E1) → (ξ(E1), τ ∗|ξ(E1)) is a homeomorphism.

Proof: (i) is clear. From (ii), the map ξ : E1 → ξ(E1) is bijective. If the map

ξ : (E1, τ∗|E1) → (ξ(E1), τ

∗|ξ(E1))

is continuous then ξ : (E1, τ ∗|E1) → (ξ(E1), τ ∗|ξ(E1)) is a homeomorphism, because (E1, τ ∗|E1)is compact and (ξ(E1), τ ∗|ξ(E1)) is Hausdorff (cf. Lemma 6.18). So, it remains to prove (ii) andthe continuity of ξ.

Continuity: Define the linear map ξ : (US/F )∗ → W∗1 by

ξ(u∗)(Φ) := −u∗(uΦ).

See Definition 3.26 for the definition of uΦ ∈ US/F . Then, for all Φ ∈ W1,

pΦ(ξ(u∗)) := |ξ(u∗)(Φ)| = |u∗(uΦ)| =: puΦ(u

∗)

for all u∗ ∈ (US/F )∗. By Theorem 6.25 the map ξ : ((US/F )∗, τ ∗) → (W∗1 , τ

∗) is continuous.Therefore, as ξ = ξ|E1 ,

ξ : (E1, τ∗|E1) → (ξ(E1), τ

∗|ξ(E1))

is continuous.

Isometry:

ξ(ρ)− ξ(ρ)= sup|ξ(ρ)(Φ)− ξ(ρ)(Φ)| : Φ ∈ W1, Φ ≤ 1= sup|(ρ− ρ)(uΦ)| : Φ ∈ W1, Φ ≤ 1≤ sup|(ρ− ρ)(A)| : A ∈ US/F , A ≤ 1.

Thenξ(ρ)− ξ(ρ) ≤ ρ− ρ.

By Lemma 1.61(i),

ρ− ρ= sup|(ρ− ρ∗ ∈ US/F , A ≤ 1= sup|(ρ− ρ∗ ∈ US/F

0 , A ≤ 1.57

For all A = A∗ ∈ UΛ, Λ ∈ Pf , we define ΦA ∈ W1 by: ΦA(Λ) := αx(A) if Λ = x + Λ forx ∈ Zd and ΦA(Λ) := 0 otherwise. Per construction ΦA = A and

ρ(A) = ρ(uΦA) = eΦA(ρ)

for all ρ ∈ E1. (Note that A = uΦA). Consequently

ρ− ρ≤ sup|eΦ(ρ)− eΦ(ρ

)| : Φ ∈ W1, Φ ≤ 1≤ sup|ξ(ρ)(Φ)− ξ(ρ)(Φ)| : Φ ∈ W1, Φ ≤ 1= ξ(ρ)− ξ(ρ).

4.3.2 The pressure as the Legendre–Fenchel transform of the free–energy density

Definition 4.12We identify via the injective map ξ : E1 → W∗

1 the set E1 with ξ(E1). In other words, anystate ρ is seen as a functional ξ(ρ) ∈ W∗

1 . For Φ ∈ W1, we then define the free–energy densityfunctional fΦ : W∗

1 → R∪∞ by fΦ(ρ) := fΦ(ρ) if ρ ∈ E1 ≡ ξ(E1) and fΦ(ρ) := ∞ otherwise.

Remark 4.13Per construction

MΦ ≡ ξ(MΦ) = u∗ ∈ W∗1 : fΦ(u

∗) = inf fΦ(W∗1 ).

Lemma 4.14fΦ : W∗

1 → R is an l.s.c. convex functional.

Proof: Clear from Theorem 4.11.

Remark 4.15 (Pressure as the Legendre–Fenchel transform)Using the dual pair ((W∗

1 , τ∗), (W1, · )) and the map fΦ from W∗

1 to R ∪ ∞ with Φ ∈ W1

(Definition 4.12), one gets

P (Φ+Ψ) = f ∗Φ(Ψ) := sup

∈W∗1

(Ψ)− fΦ()

for all Ψ ∈ W1, by Theorem 4.7.

Definition 4.16For all Φ ∈ W1,

TΦ := ∂P (Φ) ⊂ W∗1

is the set of all functionals on W1 which are tangent to P at Φ.

58

Remark 4.17Let Φ ∈ W1 and ω ∈ MΦ. By Theorem 4.7,

P (Φ+Ψ)− P (Φ) ≥ ω(Ψ) := ξ(ω)(Ψ).

for all Ψ ∈ W1. (Use fΦ(ρ) − fΦ+Ψ(ρ) = −eΨ, ρ ∈ MΦ.) See Definition 4.10 and Theorem4.11. In particular, MΦ ≡ ξ(MΦ) ⊂ TΦ.

Theorem 4.18For all Φ ∈ W1, TΦ = MΦ.

Proof: Observe that ρ ∈ MΦ when 0 ∈ ∂fΦ(ρ) (Fermat principle). For all convex l.s.c. func-tionals f , x∗ ∈ ∂f(x) when x ∈ ∂f ∗(x∗), see Theorem 6.55 (iii). Therefore, 0 ∈ ∂fΦ(ρ) yieldsρ ∈ ∂f ∗

Φ(0). From Remark 4.15 f ∗Φ(Ψ) = P (Φ+Ψ). Then ρ ∈ MΦ iff, for all Ψ ∈ W1,

P (Φ+Ψ)− P (Φ) ≥ ρ(Ψ).

Theorem 4.19 (Mazur)Let X be a Banach space and f : X → R be a continuous convex function. Let X0 ⊂ X be theset of all x ∈ X with the property that there is a unique continuous tangent functional to f atx. Then X0 is a dense set in X .

Remark 4.20 (Instability of phase transitions)From the continuity and convexity of the pressure Φ → P (Φ), it follows from Mazur’s theoremthat, for all Φ ∈ W1 and > 0, there is δΦ ∈ W1 such that δΦ < and |MΦ+δΦ| = 1.

5 Strong Coupling BCS–Hubbard Hamiltonian and Su-

perconductivity

5.1 Local model

Let S = ↑, ↓ because electrons have spin 1/2. The strong coupling Hubbard model is definedby the translation invariant interaction Φ = Φ(Λ)Λ ∈ W1 with, for all Λ ⊇ 0,

Φ(Λ) :=

−µ (n0,↑ + n0,↓)− h (n0,↑ − n0,↓) + 2λn0,↑n0,↓ if Λ = 0.0, otherwise.

for real parameters µ, h, λ, and γ ≥ 0, and with nx,s := c∗x,scx,s being the particle number

operator at position x and spin s. In the box2 ΛN := Z∩ [−L,L]d≥1 of volume |ΛN | = N ≥ 2it corresponds to the energy observable

UΦΛN

= −µ

x∈ΛN

(nx,↑ + nx,↓)− h

x∈ΛN

(nx,↑ − nx,↓) + 2λ

x∈ΛN

nx,↑nx,↓. (5.11)

2Without loss of generality we choose N such that L := (N1/d − 1)/2 ∈ N.59

The first term of the right hand side (r.h.s.) of (5.11) represents the strong coupling limit ofthe kinetic energy, with µ being the chemical potential of the system. The second term in ther.h.s. of (5.11) corresponds to the interaction between spins and the magnetic field h. Theone–site interaction with coupling constant λ represents the (screened) Coulomb repulsion asin the celebrated Hubbard model.

5.2 Superconductivity via long range interactions

Let the BCS interaction ΦBCS,ΦBCS

∈ W1 defined, for all Λ ⊇ 0, by

ΦBCS(Λ) :=

c0,↓c0,↑+c

∗

0,↓c∗

0,↑

2 if Λ = 0.0, otherwise.

and

ΦBCS

(Λ) :=

c0,↓c0,↑−c

∗

0,↓c∗

0,↑

2i if Λ = 0.0, otherwise.

The BCS energy observables in ΛN := Z ∩ [−L,L]d≥1 are

UΦBCSΛN

=

x∈ΛN

cx,↓cx,↑ + c∗

x,↓c∗x,↑

2

UΦ

BCSΛN

=

x∈ΛN

cx,↓cx,↑ − c∗

x,↓c∗x,↑

2i

and the BCS–type Hamiltonian is defined here by the long range interaction

− γ

|ΛN |(UΦBCS

ΛN+ iU

Φ

BCSΛN

)∗(UΦBCSΛN

+ iUΦ

BCSΛN

)

= − γ

N

x,y∈ΛN

c∗x,↑c

∗x,↓cy,↓cy,↑ = − γ

N

k,q∈Λ∗

N

a∗k,↑a

∗−k,↓aq,↓a−q,↑,

with γ > 0 and Λ∗N

being the reciprocal lattice of quasi–momenta and where aq,s is the corre-sponding annihilation operator for s ∈ ↑, ↓.The strong coupling BCS–Hubbard Hamiltonian is then defined by

UN = UΦΛN

− γ

|ΛN |(UΦBCS

ΛN+ iU

Φ

BCSΛN

)∗(UΦBCSΛN

+ iUΦ

BCSΛN

)

= −µ

x∈ΛN

(nx,↑ + nx,↓)− h

x∈ΛN

(nx,↑ − nx,↓) + 2λ

x∈ΛN

nx,↑nx,↓

− γ

N

x,y∈ΛN

c∗x,↑c

∗x,↓cy,↓cy,↑.

60

5.3 The approximating Hamiltonian method

SinceUΦBCSΛN

+ iUΦ

BCSΛN

∼ |ΛN |cin average, The first important remark is that one can guess the thermodynamics by theso-called approximating Hamiltonian method. In our case, the correct approximation of theHamiltonian UN is the c–dependent Hamiltonian

UN (c) : = UΦΛN

− γ

|ΛN |(|ΛN |c) (UΦBCS

ΛN+ iU

Φ

BCSΛN

)∗ − γ

|ΛN |(|ΛN |c) (UΦBCS

ΛN+ iU

Φ

BCSΛN

)

= −µ

x∈ΛN

(nx,↑ + nx,↓)− h

x∈ΛN

(nx,↑ − nx,↓) + 2λ

x∈ΛN

nx,↑nx,↓ (5.12)

− γ

N

x∈ΛN

(Nc) a∗

x,↑a∗x,↓ + (Nc) ax,↓ax,↑

, (5.13)

with c ∈ C. The main advantage of this Hamiltonian in comparison with UN is the fact thatit is a sum of shifts of the same local operator. For an appropriate order parameter c ∈ C, itleads to a good approximation of limit

limN→∞

1

βNln Tr

e−βUN

This can be partially seen from the inequality

γN |c|2 + UN (c)− UN =γ

N

x∈ΛN

a∗x,↑a

∗x,↓ −Nc

x∈ΛN

ax,↑ax,↓ −Nc

≥ 0.

Therefore, let

p (c) :=1

βNln Tr

e−βUN (c)

=

1

βln Tr

e−βU1(c)

=1

βln Tr

eβ(µ+h)n↑+(µ−h)n↓+γ(cc∗

↓c∗

↑+cc↑c↓)−2λn↑n↓

. (5.14)

5.4 Thermodynamics

Theorem 5.1 (Grand-canonical pressure)For any β, γ > 0 and µ,λ, h ∈ R,

p : = limN→∞

1

βNln Tr

e−βUN

= supc∈C

−γ|c|2 + p (c)

= β−1 ln 2 + µ+ supr≥0

m (r) < ∞,

where

m (r) := −γr +1

βlncosh (βh) + e−λβ cosh (βtr)

,

with tr := (µ− λ)2 + γ2r1/2.61

Proof: 1. For any ρ ∈ E1, by Theorem 2.11,

− 1

βNln Tr

e−βUN

=

1

NFβ(ωUN ,β) ≤

1

NFβ(ρ|UF

ΛN)

with

1

NFβ(ρ|UF

ΛN) =

ρUΦΛN

N− 1

βNS(ρ|UF

ΛN)

− γ

N2ρ(UΦBCS

ΛN+ iU

Φ

BCSΛN

)∗(UΦBCSΛN

+ iUΦ

BCSΛN

).

2. One easily compute that

limN→∞

ρUΦΛN

N= ρ(uΦ) = −µρ (n↑ + n↓)− hρ (n↑ − n↓) + 2λρ (n↑n↓) ,

because ρ ∈ E1, see also Lemma 3.28.

3. By Theorem 3.31,

s(ρ) := limN→∞

1

NS(ρ|UF

ΛN) = inf

N∈N

1

NS(ρ|UF

ΛN)

.

4. Observe that

1

N2ρ(UΦBCS

ΛN+ iU

Φ

BCSΛN

)∗(UΦBCSΛN

+ iUΦ

BCSΛN

)

= ρ

1

N2

x,y∈ΛN

c∗x,↑c

∗x,↓cy,↓cy,↑

= ρ (A∗

NAN)

with

AN :=1

N

x∈ΛN

αx(A) and A = c0,↓c0,↑ = c↓c↑.

By Theorem 3.15,

∆ (ρ) := limN→∞

ρ (A∗NAN) = p∞(U1)πρ(c0,↓c0,↑)Ωρ2 ∈ [|ρ(A)|2, A2].

The functional ∆ is affine and upper–semi–continuous (u.s.c.) in the weak∗–topology because

∆ (ρ) = infN∈N

ρ(A∗NAN) ,

as, by the proof of Theorem 3.15,

ρ (A∗NAN) = pN(U1)πρ(A)Ωρ2

≥ p∞(U1)pN(U1)πρ(A)Ωρ2 = p∞(U1)πρ(A)Ωρ2.

5. One thus has

− 1

βNln Tr

e−βUN

≤ inf f(E1),

62

where, for all ρ ∈ E1,f(ρ) := ρ(uΦ)− β−1s(ρ)− γ∆ (ρ) .

6. Similarly to Theorem 4.7 we use the state

ωN :=1

|ΛN |

x∈ΛN

ωN αx ∈ E1.

with ωN be the unique periodic product state constructed from the Gibbs state ωUN ,β. As E1

is weak–∗ compact, we can assume w.l.o.g. that the sequence ωNN∈N converges to ω∞ in theweak–∗ topology and, by lower semincontinuity of the functional

ρ → ρ(uΦ)− β−1s(ρ),

we get the limit

limN→∞

ωUN ,β

UΦΛN

N− 1

βNS(ωUN ,β)

= limN→∞

ωN(uΦ)− β−1s(ωN)

≥ ω∞(uΦ)− β−1s(ω∞).

Observe that the sequences ωNN∈N and ωUN ,βN∈N have the same weak∗–accumulationpoints. The operator UN is invariant w.r.t. permutations of lattice sites inside the boxesΛN which in turn implies the invariance of the state ωUN ,β ∈ E under permutations π ∈ Π suchthat π|L\ΛN

= id|L\ΛN. Therefore, ω∞ ∈ EΠ ⊂ E1. Moreover, for any x ∈ Zd\0,

limN→∞

ωUN ,β (A∗NAN) = lim

N→∞ωUN ,β(αx(c

∗0,↑c

∗0,↓)c

∗0,↑c

∗0,↓)

= ω∞(αx(c∗0,↑c

∗0,↓)c

∗0,↑c

∗0,↓) = ∆ (ω∞) .

It follows that

− limN→∞

1

βNln Tr

e−βUN

≥ f(ω∞) ≥ inf f(E1).

7. Thus,p = − inf f(E1)

where, for γ ≥ 0, the functional

ρ → f(ρ) := ρ(uΦ)− β−1s(ρ)− γ∆ (ρ)

is affine and lower–semi–continuous (u.s.c.) in the weak∗–topology. Since E1 is a weak∗–compactand convex set, using the Bauer maximum principle we obtain that

p = − inf f(E (E1))

with E (E1) being the set of extreme (and thus ergodic) points of E1. By Definition, theergodicity of ρ ∈ E1 implies that

∆ (ρ) := limN→∞

ρ (A∗NAN) = |ρ(A)|2.

63

Therefore,

p = − infρ∈E(E1)

−µρ (n↑ + n↓)− hρ (n↑ − n↓)

+2λρ (n↑n↓)− β−1s(ρ)− γ|ρ(c↓c↑)|2.

8. Now observe that

|ρ(c↓c↑)|2 = supc∈C

−|c|2 + 2Reρ(c↓c↑)c

,

with unique maximizer c (ρ) = ρ(c↓c↑). Therefore,

p = supc∈C

−γ|c|2 + p (c)

since

p (c) = − infρ∈E(E1)

−µρ (n↑ + n↓)− hρ (n↑ − n↓)− β−1s(ρ)

= − inf fΦ(E1)

by Theorem 4.7.

Corollary 5.2

p = − inf f(EΠ) = − inf f(E (EΠ))

where extreme states of EΠ are product states and conversely, i.e., E(EΠ) = E⊗.

5.5 Equilibrium States

Since ωUN ,βN∈N converges to a minimizer ω∞ ∈ EΠ ⊂ E1, we study the set of equilibriumstates in EΠ, i.e., the set

MΦ := ω ∈ EΠ : f (ω) = inf fΦ(EΠ)

of minimizers of the functional f . This set is a face and it suffices to know its extreme points,which are product states. In particular, since

p = supc∈C

−γ|c|2 + p (c)

= −γ|cβ|2 + p (cβ)

with cβ := r1/2β

eiφ for φ ∈ [0, 2π) and any solution rβ ∈ max 0, 1/4 of

supr≥0

m (r) = m (rβ) ,

we have, for any ω ∈ MΦ ∩ E(EΠ),

|ω(c↓c↑)|2 = −|cβ|2 + 2Reω(c↓c↑)cβ,64

and we only need to characterize the set

G :=ζ ∈ E0 : g (ζ, cβ) = −p (cβ)

with

g (ζ, c) : = −µζ (n↑ + n↓)− hζ (n↑ − n↓) + 2λζ (n↑n↓)

−β−1S(ζ)− 2γ Reζ(c↓c↑)cβ.

By Theorem 2.11, one obtain the following statement:

Theorem 5.3

G =ζcβ ∈ E0 : sup

c∈C

−γ|c|2 + p (c)

= −γ|cβ|2 + p (cβ)

with

ζc(·) :=

Tr· e−βU1(c)

Tr (e−βU1(c)). (5.15)

In particular, the extreme equilibrium states ωζcβsolves the gap equation

ωζcβ(cx,↑, cx,↓) = cβ

for any x ∈ Zd. Note that cβ := r1/2β

eiφ with rβ being equal either to zero or rβ > 0. A criticalpoint (β, γ, µ,λ, h) is when rβ > 0 and 0 are both solution of the variational problem. Notealso that the existence of several extreme equilibrium states ωζcβ

yields several (non-unitary)

irreducible GNS representations to characterize this thermodynamics. The fock space is thusnot really appropriate to describe such a behavior. Since ωUN ,β is gauge invariant, it is easynow to found its weak–∗ limit:

Corollary 5.4(i) Away from any critical point, ωUN ,β converges in the weak∗–topology to

ω∞ (·) = 1

2π

2π

0

ωζcβ(·) dφ. (5.16)

(ii) For each weak∗ limit point ω∞ of local Gibbs states ωN with parameters (βN, γ

N, µ

N,λN , hN)

converging to any critical point (β, γ, µ,λ, h), there is τ ∈ [0, 1] such that

ω∞ (·) = (1− τ)ωζ0(·) + τ

2π

2π

0

ωζcβ(·) dφ.

65

It follows that

limN→∞

ωUN ,β(c∗y,↓c

∗y,↑cx,↑cx,↓) = ω∞(c∗

y,↓c∗y,↑cx,↑cx,↓) = rβ.

If rβ > 0 then we have an (off–diagonal) long range order. In this case, we also have abreakdown of the U(1)–gauge symmetry as ωUN ,β(cx,↑cx,↓) = 0, but ωζcβ

(cx,↑, cx,↓) = cβ. The

superconducting phase is characterized by the region where rβ > 0. The Cooper pair condensatedensity equals

limN→∞

1

NωUN ,β (c

∗0c0)

= lim

N→∞

1

N2

x,y∈ΛN

ωUN ,β

a∗x,↑a

∗x,↓ay,↓ay,↑

= rβ ≤ max 0, 1/4 ,

where

c0 :=1√N

x∈ΛN

cx,↓cx,↑ =1√N

k∈Λ∗

N

ak,↓a−k,↑

resp. c∗0 annihilates resp. creates one Cooper pair within the condensate, i.e., in the zero–modefor electron pairs.

Remark 5.5The extreme equilibrium states ωζ

r1/2β

eiφφ∈[0,2π] ⊂ MΦ ∩ E(EΠ) must have irreducible rep-

resentations. Furthermore, if rβ > 0 then for any φ1,φ2 ∈ [0, 2π], φ1 = φ2 implies thatωζ

r1/2β

eiφ1= ωζ

r1/2β

eiφ2and their GNS representation cannot be (unitarly) equivalent. It is also

not unitarly equilvalent to the Fock space unless rβ = 0.

6 Appendix

6.1 Locally convex spaces

Definition 6.1 (Metric space)A pair (X , d) constructed from a set X and a real function (i.e., a metric) d : X ×X → R+

0 isa metric space if :(i) d(x, y) = 0 iff x = y.(ii) d(x, y) = d(y, x), for all x, y ∈ X .(iii) d(x, z) ≤ d(x, y) + d(y, z), for all x, y, z ∈ X .

Definition 6.2 (Topological space)A topological space is a pair (X , τ) constructed from a set X and a set τ ⊂ 2X such that:(i) ∅,X ∈ τ .(ii) For all U1, . . . , UN ∈ τ , N ∈ N, one has U1 ∩ · · · ∩ UN ∈ τ .(iii) For all Ui, i ∈ I with I being any index set, there is

i∈I

Ui ∈ τ .

66

Such a τ is called a topology of X . Element U ∈ τ are known as open sets of X . In particular,V ⊂ X is closed if X\V ∈ τ .For all x ∈ X , Ux ⊂ X is a neighborhood of x if there is an open set V ∈ τ such thatx ∈ V ⊂ Ux.Let X be a set and τ , τ two topologies of X . τ is finer than τ if τ ⊃ τ (in this case, τ iscoarser than τ).

Definition 6.3 (Topological metric space)Let (X , d) be a metric space. A subset U ⊂ X is open if, for all x ∈ U , there is > 0 such that

B(x) := y ∈ X : d(x, y) < ⊂ U.

The setτ d := U ⊂ X : U open

is called the topology of the metric space (X , d). A topological space (X , τ) is said to be metriz-able if there is a metric d : X × X → R+

0 such that τ d = τ .

Lemma 6.4If (X , d) is a metric space then (X , τ d) is a topological space (in the sense of Definition 6.2).

Definition 6.5 (Relative topology)Let (X , τ) be a topological space and Y ⊂ X . Then

τ |Y := Y ∩ U : U ∈ τ

is called the relative topology of Y with respect to τ .

Definition 6.6 (Product topology)Let (X1, τ 1) and (X2, τ 2) be topological spaces. A subset U ⊂ X1×X2 is called open in the producttopology if, for any (x1, x2) ∈ U , there is a neighborhood Ux1 of x1 ∈ X1 and a neighborhoodUx2 of x2 ∈ X2 such that Ux1 × Ux2 ⊂ U . The set

τ 1 × τ 2 := U ⊂ X1 × X2 : U open (product topology)

is called the product topology of (X1, τ 1) and (X2, τ 2).

Lemma 6.7Let (X , τ) and (X , τ ) be topological spaces and Y ⊂ X . Then (X × X , τ × τ ) and (Y , τ |Y)are both topological spaces (in the sense of Definition 6.2).

Definition 6.8 (Hausdorff space)A topological space (X , τ) is a Hausdorff space if, for any x, y ∈ X , x = y, there is open setsUx, Uy ∈ τ such that x ∈ Ux, y ∈ Uy, and Ux ∩ Uy = ∅. In particular, metric spaces areHausdorff spaces.

67

Definition 6.9 (Continuous maps)Let (X , τ) and (Y , τ ) be topological spaces. The map f : X → Y is continuous if one of thefollowing equivalent assertions holds:(i) For all open sets U ⊂ Y, the set f−1(U) ⊂ X is open.(ii) For all closed sets U ⊂ Y, the set f−1(U) ⊂ X is closed.(iii) For any x ∈ X and any neighborhood Uf(x) ⊂ Y of f(x) ∈ Y, f−1(Uf(x)) is a neighborhoodof x.

Definition 6.10 (Homeomorphisms)A homeomorphism is a bijective continuous map f : X → Y such that f−1 is also continuous.

Definition 6.11 (Topological vector space)Let either K = C or K = R and X be a K–vector space. If τ is a topology of X such that themaps X × X → X , (x, y) → x + y (addition), and K × X → X , (λ, x) → λx, are continuous(in the product topology), then (X , τ) is called a topological vector space.

Definition 6.12 (Locally convex vector space)A Hausdorff topological vector space (X , τ) is a locally convex vector space (LCV) if one of thefollowing equivalent assertions holds:(i) For all neighborhood U0 of 0 ∈ X , there is a convex neighborhood V0 of 0 such that V0 ⊂ U0.(ii) For all x ∈ X and all neighborhood Ux of x ∈ X , there is a convex neighborhood Vx of xwith Vx ⊂ Ux.

Definition 6.13 (Bases and subbases)Let (X , τ) be a topological space. The system of subsets B ⊂ 2X is called a basis of the topologyτ if, for all U ∈ τ , there is Ui ∈ B, i ∈ I, such that

U =

i∈IUi.

The system of subsets S ⊂ 2X is a subbasis of the topology τ if

U1 ∩ · · · ∩ UN : U1, . . . , UN ∈ S, N ∈ N

is a Basis of τ . Per convention, ∩i∈∅Ui := X .

Remark 6.14Two topologies with same basis are equal to each others.

Lemma 6.15Let X be a set and S ⊂ 2X any system of subsets of X . Then there is a unique topology τ(S)of X such that S is a subbasis. τ(S) is the topology constructed from S and corresponds to(arbitrary, not necessarily countable) unions of intersections of finitely many sets of S.

68

Definition 6.16 (Compacticity)Let (X , τ) be a topological space. A set K ⊂ X is compact if all open covers of K has a finitesubcover, i.e., K ⊂

i∈I

Ui, Ui ∈ τ , i ∈ I implies that there is J ⊂ I such that |J | < ∞ and

K ⊂j∈J

Uj.

Remark 6.17Let (X , τ) be a topological space.(i) Let (Y , τ ) be another topological space and f : X → Y a continuous map. Then, for allcompact sets K ⊂ X , f(K) ⊂ Y is also compact.(ii) If K ⊂ X is compact and K ⊂ K is closed then K is compact.(iii) If (X , τ) is Hausdorff and K ⊂ X is compact then K is closed.

Lemma 6.18Let (X , τ) be a compact topological space and (Y , τ ) a Hausdorff space. A continuous andbijective map f : X → Y is a homeomorphism.

Proof: Using Remark 6.17 (ii), all closed sets U ⊂ X are compact and, by Remark 6.17 (i) theset f(U) ⊂ Y is also compact. By Remark 6.17 (iii), f(U) is closed.

Definition 6.19 (Semi–norm)Let X be a K–vector space. The map p : X → R+

0 is a semi–norm if :(i) p(λx) = |λ|p(x), for all λ ∈ K and all x ∈ X .(ii) p(x+ y) ≤ p(x) + p(y), for all x, y ∈ X .

Definition 6.20 (Topology of semi–norms)Let X be a K–vector space and P = pii∈I , pi : X → R+

0 , be a family of semi–norms. For allx ∈ X , i ∈ I and all > 0, define

Ux

(i,) := y ∈ X : pi(y − x) < .

andSP := Ux

(i,) : x ∈ X , > 0, i ∈ I ⊂ 2X .

Then the set τP := τ(SP ) is the topology produced by the system P of semi–norms. For allx ∈ X , i1, . . . , in ∈ I, 1, . . . , n > 0, and all n ∈ N, the corresponding (open) neighborhood ofx equals

Ux

(i1,1),...,(in,n) := Ux

(i1,1) ∩ · · ·Ux

(in,n) ∈ τP .

[cf. Definition 1.28]

Theorem 6.21Let X be a K–vector space and P = pii∈I , pi : X → R+

0 , be a family of semi–norms. Thenone has:(i) If

i∈I, >0

U0(i,) = 0, (6.17)

69

then (X , τP ) is a locally convex vector space (LCV). In particular, (X , τP ) is Hausdorff.(ii) Conversely, if (X , τ) is a locally convex vector space then there is a system P of semi–normssuch that τ = τP and (6.17) holds.

Example 6.22 Every normed space (X , · ) is a LCV with respect to (w.r.t.) the topology ofthe norm: τ = τP , with P = p and p(x) = x.

Example 6.23Let X be a Banach space and X ∗ be the space of all continuous linear functionals from X to K.For all x ∈ X , let px : X ∗ → R+

0 be the semi–norm defined by px(x∗) = |x∗(x)|. If τ ∗ = τP andP = pxx∈X then (X ∗, τ ∗) is a LCV. τ ∗ is called the weak∗–topology of X ∗.

Theorem 6.24 (Metrizable LCV)Let X be a K–vector space and P = pnn∈N be a countable family of semi–norms such that

n∈N, >0

U0(n,) = 0.

Then the map d : X × X → R+0 defined by

d(x, y) :=

k∈N2−k

pk(x− y)

1 + pk(x− y),

for all x, y ∈ X , defines a metric on X such that τ d = τP . All metrizable LCV are of this form.

Theorem 6.25 (Cont. Linear. funct in LCV)Let (X , τ) and (Y , τ ) be two LCV. Let P = pii∈I and P = p

jj∈J be two systems of semi–

norms such that τ = τP and τ = τP . A linear map f : X → Y is continuous if, for any j ∈ J ,there are a finite number of i1, . . . , in ∈ I and a constant Cj < ∞ such that

pj(f(x)) ≤ Cj maxpi1(x), . . . , pin(x), ∀x ∈ X .

Corollary 6.26Let X be a Banach space and X ∗ be the space of all continuous linear functionals from X to K.Then one has:(i) The topology of the norm of X ∗ is finer than the weak∗–topology of X ∗.(ii) The evaluation of the functional x∗ ∈ X ∗ at some fixed point x ∈ X , i.e., the map x∗ →x∗(x) ∈ K, is continuous w.r.t. the weak∗–topology.

Proof: (i) It is sufficient to prove that the map id : (X ∗, · ) → (X ∗, τ ∗), id(x) := x, iscontinuous. Using the definition of the norm of X ∗, for all x ∈ X ,

px(x∗) = |x∗(x)| ≤ xXx∗X ∗ = Cx∗X ∗

For all x∗ ∈ X ∗. Using Theorem 6.25 it follows that id : (X ∗, · ) → (X ∗, τ ∗) is continuous.

(ii) is proven in the same way as

|x∗(x)| ≤ 1 · px(x∗), ∀x∗ ∈ X ∗,

for any x ∈ X .

70

Definition 6.27Let X be a C∗–algebra and α = αii∈I be a family of automorphisms of X . The state ω ∈ X ∗

+,1

is symmetric (w.r.t. α) if, for all i ∈ I, ω αi = ω. α represents the physical symmetry of thesystem S with set O = OX of observables. The set of all symmetric states of X ∗

+,1 are denotedby X ∗

+,1,α.

Corollary 6.28Let X be a C∗–algebra and α = αii∈I be a family of automorphisms of X . Then X ∗

+,1,α isconvex and weak∗–compact.

Proof: Because of the linearity of states,

(λω + (1− λ)ω) αi = λω αi + (1− λ)ω αi,

for all ω,ω ∈ X ∗+,1, λ ∈ [0, 1] and i ∈ I. In particular, X ∗

+,1,α is convex. Per definition andlinearity of states,

X ∗+,1,α

= X ∗+,1

A∈X , i∈Iω ∈ X ∗

+,1 : ω(A− αi(A)) = 0.

Since 0 is a closed set of R, by weak∗–continuity of the map x∗ → x∗(x) ∈ K (Corollary 6.26(ii)), the set

ω ∈ X ∗+,1 : ω(A− αi(A)) = 0

is weak∗–closed. Therefore, X ∗+,1,α ⊂ X ∗

+,1 is weak∗–closed and then weak∗–compact, by weak∗–

compacticity of X ∗+,1.

Theorem 6.29Let X be a separable Banach space, X0 = xnn∈N be a dense subset of X , and P0 = pn =pxnn∈N be a family of semi–norms. Define

B∗1(0) := x∗ ∈ X ∗ : x∗ ≤ 1.

Thenτ ∗|B∗

1 (0)= τP0 |B∗

1 (0).

(see Example 6.23). Note that, in general, τ ∗ = τP0.

Proof: By density of the set X0, for any (non–zero) x∗ ∈ X ∗, there is n ∈ N such thatpn = |x∗(xn)| > 0. In particular,

n∈N, >0

U0(n,) = 0.

From Theorem 6.24, (X ∗, τP0) is a metrizable LCV. Since, for all n ∈ N and x∗ ∈ X ∗,

pn(x∗) ≤ px(x

∗)71

with x = xn, the identity mapid : (X ∗, τ ∗) → (X ∗, τP0)

is continuous, by Theorem 6.25. In particular,

id : (B∗1(0), τ

∗|B∗1 (0)

) → (B∗1(0), τP0 |B∗

1 (0))

is continuous. Using Banach–Alaoglu’s theorem the set B∗1(0) is compact w.r.t. τ ∗, i.e.,

(B∗1(0), τ

∗|B∗1 (0)

) is a compact space. (B∗1(0), τP0 |B∗

1 (0)) is Hausdorff as (X ∗, τP0) is Hausdorff.

By Lemma ??, it follows that

id : (B∗1(0), τ

∗|B∗1 (0)

) → (B∗1(0), τP0 |B∗

1 (0))

is a homeomorphism.

Corollary 6.30Let X be a separable C∗–algebra and α = αii∈I be a family of automorphisms of X . Then(X ∗

+,1, τ∗|X ∗

+,1) and (X ∗

+,1,α, τ∗|X ∗

+,1,α) are convex, compact, and metrizable.

Proof: From Corollary 6.28, X ∗+,1 and X ∗

+,1,α are convex and compact w.r.t. τ ∗. Let P0 be acountable system of semi–norms. By Theorem 6.29, the relative topologies of X ∗

+,1 and X ∗+,1,α

in (X ∗, τ ∗) and (X ∗, τP0) are equal to each others because ω = 1 for all ω ∈ X ∗+,1. Using

Theorem 6.24, the topological space (X ∗, τP0) is metrizable.

6.2 Lower–semi–continuous functionals

Definition 6.31Let (X , τ) be a topological vector space. A functional f : X → R ∪ ∞ is lower–semi–continuous (l.s.c.) if one of these equivalent conditions holds:(i) For all c ∈ R, the set f−1((−∞, c]) is closed.(ii) For all c ∈ R, the set f−1((c,∞]) is open.A functional f : X → R ∪ −∞ is upper–semi–continuous (u.s.c.) if −f is lower–semi–continuous.

Lemma 6.32Let (X , d) be a metrizable space and f : X → R ∪ ∞ be a l.s.c. functional. Then, for allconvergence sequences xn → x in X ,

f(x) ≤ lim infn→∞

f(xn).

Proof: Let x ∈ X and xn → x be a convergence sequence. Since f is l.s.c., for all > 0,

V := f−1((f(x)− ,∞])

is an open neighborhood of x. In particular, for all > 0 there is η> 0 such that f(x) ≥

f(x)− for all x ∈ Bη(x). Then

lim infn→∞

f(xn) ≥ f(x)−

for all > 0.

72

6.3 Compact convex sets and Choquet simplices

The theory of compact convex subsets of a locally convex Hausdorff (topological vector) realspace X is standard. For more details, see, e.g., [2, 3].

One important observation concerning locally convex Hausdorff real spaces X is that anycompact convex subset K ⊂ X is the closure of the convex hull of the (non–empty) set E(K)of its extreme points, i.e., of the points which cannot be written as – non–trivial – convexcombinations of other elements in K. This is Krein–Milman theorem (see, e.g., [1, Theorems3.4 (b) and 3.23]):

Theorem 6.33 (Krein–Milman)Let K ⊂ X be any (non–empty) compact convex subset of a locally convex Hausdorff real spaceX . Then we have that:(i) The set E(K) of its extreme points is non–empty.(ii) The set K is the closed convex hull of E(K).

Remark 6.34 X being a Hausdorff topological vector space on which its dual space X ∗ sepa-rates points is the only condition necessary on X in Krein–Milman theorem. For more details,see, e.g., [1, Theorem 3.23].

In fact, the set E(K) of extreme points is even a Gδ set if the compact convex set K ⊂ Xis metrizable. Moreover, among all subsets Z ⊂ K generating K, E(K) is – in a sense – thesmallest one (see, e.g., [3, Proposition 1.5]):

Theorem 6.35 (Properties of the set E(K))Let K ⊂ X be any (non–empty) compact convex subset of a locally convex Hausdorff real spaceX . Then we have that:(i) If K is metrizable then the set E(K) of extreme points of K forms a Gδ set.(ii) If K is the closed convex hull of Z ⊂ K then E(K) is included in the closure of Z.

Property (i) can be found in [3, Proposition 1.3] and only needs that X is a topological vec-tor space, whereas the second statement (ii) is a classical result obtained by Milman, see [3,Proposition 1.5].

Theorem 1.31 restricted to finite dimensions is a classical result of Minkowski which, for anyx ∈ K in (non–empty) compact convex subset K ⊂ X , states the existence of a finite numberof extreme points x1, . . . , xk ∈ E(K) and positive numbers µ1, . . . , µk

≥ 0 with Σk

j=1µj= 1 such

that

x =k

j=1

µjxj. (6.18)

To this simple decomposition we can associate a probability measure, i.e., a normalized positiveBorel regular measure, µ on K.

73

Indeed, the Borel sets of any set K are elements of the σ–algebra B generated by closed –or open – subsets of K. Positive Borel regular measures are the positive countably additive setfunctions µ over B satisfying

µ (B) = sup µ (C) : C ⊂ B, C closed = inf µ (O) : B ⊂ O, O open

for any Borel subset B ∈ B of K. If K is compact then any positive Borel regular measure µcorresponds (one–to–one) to an element of the setM+(K) of Radon measures with µ (K) = µand we write

µ (h) =

K

dµ(x) h (x) (6.19)

for any continuous function h on K. A probability measure µ ∈ M+1 (K) is per definition a

positive Borel regular measure µ ∈ M+(K) which is normalized : µ = 1.

Remark 6.36 The set M+1 (K) of probability measures on K can also be seen as the set of

states on the commutative C∗–algebra C(K) of continuous functionals on the compact set K,by the Riesz–Markov theorem.

Therefore, using the probability measure µx∈ M+

1 (K) on K defined by

µx=

k

j=1

µjδxj

with δy being the Dirac – or point – mass3 at y, Equation (6.18) can be seen as an integraldefined by (6.19) for the probability measure µ

x∈ M+

1 (K):

x =

K

dµx(x) x . (6.20)

The point x is in fact the barycenter of the probability measure µx. This notion is defined in

the general case as follows (cf. [3, p. 1]):

Definition 6.37 (Barycenters of probability measures in convex sets) Let K ⊂ X beany (non–empty) compact convex subset of a locally convex real space X and let µ ∈ M+

1 (K)be a probability measure on K. We say that x ∈ K is the barycenter4 of µ if, for all continuouslinear5 functionals h on X ,

h (x) =

K

dµ(x) h (x) .

Barycenters are well–defined for all probability measures in convex compact subsets of locallyconvex spaces (cf. [3, Propositions 1.1 and 1.2]):

3δy is the Borel measure such that for any Borel subset B ∈ B of K, δy(B) = 1 if y ∈ B and δy(B) = 0 ify /∈ B.

4Other terminology existing in the literature: “x is represented by µ”, “x is the resultant of µ”.5Barycenters can also be defined in the same way via affine functionals instead of linear functionals, see [8,

Proposition 4.1.1.].

74

Theorem 6.38 (Well-definiteness and uniqueness of barycenters)Let K ⊂ X be any (non–empty) compact subset of a locally convex real space X such that co (K)is also compact. Then we have that:(i) For any probability measure µ ∈ M+

1 (K) on K, there is a unique barycenter xµ ∈ co (K).In particular, if K is convex then, for any µ ∈ M+

1 (K), there is a unique barycenter xµ ∈ K.Moreover, the map µ → xµ from M+

1 (K) to co (K) is affine and weak∗–continuous.(ii) Conversely, for any x ∈ co (K), there is a probability measure µ

x∈ M+

1 (K) on K withbarycenter x.

Therefore, we write the barycenter xµ of any probability measure µ in K as

xµ =

K

dµ(x) x,

where the integral has to be understood in the weak sense. By Definition 6.37, it means thath (xµ) can be decomposed by the probability measure µ ∈ M+

1 (K) provided h is a continuouslinear functional. In fact, this last property can also be extended to all affine upper semi–continuous functionals on K, see, e.g., [8, Corollary 4.1.18.] together with [1, Theorem 1.12]:

Lemma 6.39 (Barycenters and affine maps)Let K ⊂ X be any (non–empty) compact convex subset of a locally convex Hausdorff real spaceX . Then, for any probability measure µ ∈ M+

1 (K) on K with barycenter xµ ∈ K and for anyaffine upper semi–continuous functional h on K,

h (x) =

K

dµx(x) h (x) .

It is natural to ask whether, for any x ∈ K in a convex set K, there is a (possibly not unique)probability measure µ

xon K supported on E(K) with barycenter x. Equation (6.20) already

gives a first positive answer to that problem in the finite dimensional case. The general case hasbeen proven by Choquet, whose theorem is a remarkable refinement of Krein–Milman theorem(see, e.g., [3, p. 14]):

Theorem 6.40 (Choquet)Let K ⊂ X be any (non–empty) metrizable compact convex subset of a locally convex Hausdorffreal space X . Then, for any x ∈ K, there is a probability measure µ

x∈ M+

1 (K) on K such that

µx(E(K)) = 1 and x =

K

dµx(x) x.

Recall that the integral above means that x ∈ K is the barycenter of µx.

Remark 6.41 (Choquet theorem and affine maps)By Lemma 6.39, Choquet theorem can be used to decompose any affine upper semi–continuousfunctional defined on the metrizable compact convex subset K ⊂ X w.r.t. extreme points of K.

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Remark 6.42 (Choquet theorem for non metrizable K)If the (non–empty) compact convex subset K ⊂ X is not metrizable then E(K) may not forma Borel set. Choquet theorem (Theorem 6.40) stays, however, valid under the modification thatµxis pseudo–supported by E(K) which means that µ

x(B) = 1 for all Baire sets B ⊇ E(K).

This result is known as the Choquet–Bishop–de Leeuw theorem, see [3, p. 17].

Note that the probability measure µxof Theorem 6.40 is a priori not unique. For instance, in

the 2-dimensional plane, simplices (points, segments, and triangles) are uniquely decomposedin terms of their extreme points, i.e., they are uniquely represented by a convex combination ofextreme points. But this decomposition is not anymore unique for a square. In fact, uniquenessof the decomposition given in Theorem 6.40 is related to the theory of simplices.

To define them in the general case, let S be a compact convex set of a locally convex Hausdorffreal space X . Without loss of generality assume that the compact convex set S is included ina closed hyper–plane which does not contain the origin6. Let

K := αx : α ≥ 0, x ∈ S

be the cone with base S. Recall that the cone K induces a partial ordering on X by using thedefinition x y iff x − y ∈ K. A least upper bound for x and y is an element x ∨ y x, ysatisfying w x ∨ y for all w with w x, y. Then a simplex is defined as follows:

Definition 6.43 (Simplices)The (non–empty) compact convex set S is a simplex whenever K is a lattice with respect to thepartial ordering . This means that each pair x, y ∈ K has a least upper bound x ∨ y ∈ K.

Observe that a simplex can also be defined for non–compact convex sets but we are onlyinterested here in compact simplices. Such simplices are particular examples of simplexoids,i.e., compact convex sets whose closed proper faces are simplices.

The definition of simplices above agrees with the usual definition in finite dimensions as the n–dimensional simplex (λ1,λ2, · · · ,λn+1, ),Σjλj = 1 is the base of the (n+1)–dimensional cone(λ1,λ2, · · · ,λn+1, ),λj ≥ 0. In fact, for all metrizable simplices, the probability measure µ

xof

Theorem 6.40 is unique and conversely, if µxis always uniquely defined then the corresponding

metrizable compact convex set is a simplex (see, e.g., [3, p. 60]):

Theorem 6.44 (Choquet)Let S ⊂ X be any (non–empty) closed convex metrizable subset of a locally convex space X .Then S is a simplex iff, for any x ∈ S, there is a unique probability measure µ

x∈ M+

1 (S) onS such that

µx(E(S)) = 1 and x =

S

dµx(x) x.

Compact and metrizable convex sets for which the integral representation in Theorem 6.44 isunique are also called Choquet simplices :

6Otherwise, we embed X as X × 1 in X × R.76

Definition 6.45 (Choquet simplex)The simplex S is a Choquet simplex whenever the decomposition of S on E(S) given by Theorem6.40 (see also Remark 6.42) is unique.

Two further special types of simplices are of particular importance: The Bauer and thePoulsen simplices. The first one is defined as follows:

Definition 6.46 (Bauer simplex)The simplex S is a Bauer simplex whenever its set E(S) of extreme points is closed.

A compact Bauer simplex S has the interesting property that it is affinely homeomorphic tothe set of states on the commutative C∗–algebra C(E(S)) (see, e.g., [2, Corollary II.4.2]):

Theorem 6.47 (Bauer)Let S ⊂ X be any compact Bauer Simplex of a locally convex Hausdorff real space X . Then themap x → µ

xdefined by Theorem 6.44 from S to the set M+

1 (E(S)) of probability measures7 onE(S) is an affine homeomorphism.

Bauer simplices are special simplices as the set of E(S) of extreme points of a simplex Smay not be closed. In fact, E. T. Poulsen [15] constructed in 1961 an example of a metrizablesimplex S with E(S) being dense in S. This simplex is now well–known as the Poulsen simplexbecause it is unique [13, Theorem 2.3.] up to an affine homeomorphism:

Theorem 6.48 (Lindenstrauss–Olsen–Sternfeld)Every (non–empty) compact metrizable simplex S with E(S) being dense in S is affinely home-omorphic to the Poulsen simplex.

The original example given by Poulsen [15] is not explained here as we give in Section ?? aprototype of the Poulsen simplex: The set E1 ⊂ U∗ of all t.i. states, see Theorem 3.21.

For more details on the Poulsen simplex we recommend [13] where its specific properties aredescribed. They also show that the Poulsen simplex is, in a sense, complementary to the Bauersimplices, see [2, p. 164] or [13, Section 5].

6.4 The Legendre–Fenchel transform

Definition 6.49 (Dual pairs)Let (X , τ) be a LCV and X ∗ be the set of all continuous (w.r.t. the topology τ) linear functionalsx∗ : X → K. Let τ be a topology of X ∗ such that:(i) (X ∗, τ ) is a LCV.(ii) For all x ∈ X , the linear map, x∗ → x∗(x) is continuous w.r.t. τ .(iii) Any continuous (w.r.t. τ ) linear functional ξ : X ∗ → K has the form ξ(x∗) = x∗(x) forsome fixed x ∈ X and all x∗ ∈ X ∗.Such a ((X , τ), (X ∗, τ )) is called a dual pair.

7I.e. the set of states on the commutative C∗–algebra C(E(S)) of continuous functionals on the compact setE(S).

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Lemma 6.50Let X be a Banach space. For any continuous map f : (X ∗, τ ∗) → K, there is x ∈ X such thatf(x∗) = x∗(x) for all x∗ ∈ X ∗. In particular, ((X , · ), (X ∗, τ ∗)) is a dual pair.

Proof: Assume w.l.o.g. that Ker f = X . Otherwise f(x∗) = 0 = x∗(0) for all x∗ ∈ X ∗.

(a) Let ξ1, . . . , ξm : X → K be m ∈ N linearly independent maps. Then, there is x1, . . . , xm ∈X such that ξ

k(xl) = δkl for k, l = 1, 2, . . . ,m. This holds for m = 1. Therefore the

case m > 1 is shown by induction: Let x1, . . . , x

m−1 ∈ X such that ξ

k(x

l) = δkl for

k, l = 1, 2, . . . ,m− 1. Define

Xm := x ∈ X : ξ1(x) = · · · = ξm−1(x) = 0.

Then ξm|Xm = 0: Indeed, assume that ξ

m|Xm = 0. Define the linear functional

ξm(x) := ξ

m(x

1)ξ1(x) + . . .+ ξm(x

m−1)ξm−1(x).

Observe that, for all x ∈ X ,

ξ1(x)x1 + . . .+ ξ

m−1(x)xm−1 − x ∈ Xm.

From ξn|Xm = 0, we obtain

(ξm− ξ

m)(x)

= ξm(x+ (ξ1(x)x

1 + . . .+ ξ

m−1(x)xm−1 − x))

−ξm(x

1)ξ1(x)− . . .− ξm(x

m−1)ξm−1(x)

= 0

for all x ∈ X . This means that ξm= ξ

mand ξ

mis linear combination of ξ1, . . . , ξm−1.

Therefore, there is xm ∈ Xm such that ξm(xm) = 1. Define, for k = 1, 2, . . . ,m− 1,

xk := xk− ξ

m(x

k)xm.

Then ξk(xl) = δk,l for all k, l = 1, 2, . . . ,m.

(b) Let ξ : X ∗ → K be a linear and continuous (w.r.t. τ ∗) functional. Then, by Theorem6.25, there are x1, . . . , xn ∈ X , n ∈ N, and a constant C < ∞ such that, for all x∗ ∈ X ∗,

|ξ(x∗)| ≤ Cmax|x∗(x1)|, . . . , |x∗(xn)|.

In particular,

Ker ξ ⊃n

k=1

x∗ ∈ X ∗ : x∗(xk) = 0. (6.21)

(c) From (a) and (b) it follows that ξ is a linear combination of functionals ξk(x∗) := x∗(xk)

with k = 1, 2, . . . , n and xk be defined as in (b): If ξ0, ξ1, . . . , ξn are linearly independentwith ξ0 := ξ, then there is x0 ∈ X such that ξ0(x0) = δ0,0 = 1 but ξ

k(x0) = δ0,k = 0 for

all k = 1, 2, . . . , n. This contradicts (6.21).

78

Remark 6.51(i) If ((X , τ), (X ∗, τ )) is a dual pair then ((X ∗, τ ), (X , τ)) is also a dual pair (see definition).(ii) If (X , ·) is a Banach space then the weak∗–topology τ ∗ is the biggest local convex topologyof X ∗ such that ((X , τ), (X ∗, τ )) is a dual pair.

Definition 6.52 (Legendre–Fenchel transform)Let ((X , τ), (X ∗, τ )) be a dual pair. For any functional f : X → R ∪ ∞, f = ∞, and allx∗ ∈ X ∗, its Legendre–Fenchel transform is the map f ∗ : X ∗ → R ∪ ∞ defined by

f ∗(x∗) := supx∈X

x∗(x)− f(x) ∈ R ∪ ∞.

for x∗ ∈ X ∗.

Lemma 6.53Let ((X , τ), (X ∗, τ )) be a dual pair and a function f : X → R ∪ ∞, f = ∞.(i) For all y ∈ X , let fy : X → R ∪ ∞ be defined by fy(x) := f(x − y). Then f ∗

y(x∗) =

f ∗(x∗) + x∗(y).(ii) The Legendre–Fenchel transform f ∗ : (X ∗, τ ) → R∪ ∞ of f : X → R∪ ∞ is a convexl.s.c. function.

Proof: (i) result from the definition. For (ii) see proof of Theorems 2.14 and 3.33.

Definition 6.54Let ((X , τ), (X ∗, τ )) be a dual pair.(i) Γ(X , τ) is the set of all convex l.s.c. (w.r.t. τ) functionals f : X → R ∪ ∞, f = ∞.(ii) For all f ∈ Γ(X , τ) and x ∈ X , the set

∂f(x) := x∗ ∈ X ∗ : ∀y ∈ X f(x+ y) ≥ f(x) + x∗(y).

is the set of all functionals on X which are tangent to f at x. (cf. Definition 2.15).

Theorem 6.55 (Properties of the Legendre–Fenchel transform)(i) The Legendre–Fenchel transform f → f ∗ is a bijection from Γ(X , τ) to Γ(X ∗, τ ).(ii) For all f ∈ Γ(X , τ), f ∗∗ := (f ∗)∗ = f .(iii) For all f ∈ Γ(X , τ), x∗ ∈ ∂f(x) when x ∈ ∂f ∗(x∗).

Proof: See, e.g., E. Zeidler. Nonlinear Functional Analysis and its Applications Band III.Chapter 51.

6.5 Algebras

6.5.1 Definitions

Definition 6.56 (Algebra)Let X be a complex vector space with the product map X × X → X , (A,B) → AB. X is a

79

associative and distributive algebra, when(i) ∀A,B,C ∈ X : (AB)C = A(BC).(ii) ∀A,B,C ∈ X : (A+B)C = AB +BC and C(A+B) = CA+ CB.(iii) ∀A,B ∈ X , ∀α, β ∈ C: αβ(AB) = (αA)(βB).

X is commutative if ∀A,B ∈ X , AB = BA. 1 ∈ X is the identity when A1 = 1A = A forall A ∈ X .

Definition 6.57 (∗–Algebra)The map A → A∗ from an algebra X to X is an involution, when(i) ∀A ∈ X : (A∗)∗ = A.(ii) ∀A,B ∈ X : (AB)∗ = B∗A∗.(iii) ∀A,B ∈ X , ∀α, β ∈ C: (αA+ βB)∗ = αA∗ + βB∗.An algebra X with an involution is a ∗–algebra and A ∈ X is self–adjoint (s.a.) when A = A∗.

Remark 6.58(i) If the identity 1 ∈ X exists then it is unique and if additionally X is a ∗–algebra, 1 isself–adjoint.(ii) The set

OX := A = A∗ : A ∈ X,of a ∗–algebra X , is a real vector space, see Theorem 1.16.

Definition 6.59 (Normed algebra)(i) (X , · ) is a normed algebra if X is an algebra and, for all A,B ∈ X ,

AB ≤ AB.

(ii) A normed algebra (X , · ) is a Banach algebra if X is complete (w.r.t. to the norm · ).(iii) A Banach algebra (X , · ) with an involution ∗ such that, for all A ∈ X , A = A∗, isa Banach ∗–algebra.(iv) A Banach ∗–algebra (X , · , ∗) is a C∗–algebra if, for all A ∈ X , A∗A = A2.

Remark 6.60(i) Let (X , · ) be a normed ∗–algebra. For all A,B ∈ OX ,

A •s B =1

2AB +BA ≤ AB.

see Theorem 1.39 (i).(ii) Let (X , · , ∗) be a C∗–algebra. Then, for all A ∈ OX ,

A2 = A∗A = A2,

see Theorem 1.20.(iii) Using AB ≤ AB,

AB − AB ≤ A− AB,AB − AB ≤ AB − B.

80

Then the product (A,B) → AB is also continuous.(iv) The equality A = A∗ implies the continuity of the involution:

A∗ − B∗ = A− B.

(v) If 1 ∈ OX ⊂ X is the identity of a Banach ∗–algebra then 1 = 1.

6.5.2 Spectrum in algebras

Definition 6.61 (Inverse)Let X be a algebra with identity. A ∈ X is invertible if there is A−1 ∈ X such that

AA−1 = A−1A = 1.

Lemma 6.62If the inverse A−1 of A ∈ X exists then it is unique. If A,B ∈ X are invertible, then AB isinvertible and

(AB)−1 = B−1A−1.

If X is a ∗–algebra and A ∈ X is invertible then A∗ is invertible and (A∗)−1 = (A−1)∗.

Definition 6.63 (Resolvent and spectrum)Let X be a algebra with identity. The resolvent set (A) of A is the set of all λ ∈ C such that(λ1− A) is invertible. The spectrum of A is σ(A) := C\(A). For all λ ∈ (A), the operator

R(A,λ) := (λ1− A)−1 ∈ X

is called the resolvent of A at λ.

Lemma 6.64 (Spectrum in Banach algebras)Let X be a Banach algebra with identity. Then, for all A,B ∈ X such that A is invertible withBA−1 < 1, (A− B) is invertible and

(A− B)−1 = A−1 + A−1∞

n=1

(BA−1)n.

[Neumann series]. In particular, for all A ∈ X , (A) ⊂ C is open and σ(A) ⊂ C is compact.

Theorem 6.65 (Spectral radius in Banach algebras)Let X be a Banach algebra with identity. Then the spectral radius

r(A) := sup|λ| : λ ∈ σ(A) ≥ 0

of A ∈ X equalsr(A) = inf

n∈NAn1/n = lim

n→∞An1/n ≤ A.

81

Lemma 6.66Let X be a ∗–algebra with identity. Then we have that :(i) For all A ∈ X and λ ∈ C:

σ(λ1− A) = λ− σ(A) := λ− x : x ∈ σ(A)

andσ(A∗) = σ(A) := x : x ∈ σ(A).

(ii) If A is invertible then 0 /∈ σ(A) ∪ σ(A−1) and

σ(A−1) = σ(A)−1 := x−1 : x ∈ σ(A).

(iii) For all A,B ∈ X ,σ(AB) ∪ 0 = σ(BA) ∪ 0.

Theorem 6.67 (Spectrum in C∗–algebras)Let X be a C∗–algebra with identity and A ∈ X . Then we have that :(i) If A is normal, i.e., AA∗ = A∗A, then the spectral radius of A is maximal, i.e., r(A) = A.(ii) If A is unitary, i.e., AA∗ = A∗A = 1, then,

σ(A) ⊂ λ ∈ C : |λ| = 1.

(iii) If A is self–adjoint, i.e., A = A∗, then

σX (A) ⊂ [−A, A], σX (A2) ⊂ [0, A2].

(iv) If P is a polynomial with complex coefficients, then

σ(P(A)) = P(σ(A)) := P(λ) : λ ∈ σ(A).

Corollary 6.68 (Uniqueness of the C∗–norm)Let X be a ∗–algebra. Then, there is a norm · of X such that (X , · ) is a C∗–algebra.

Proof: For A ∈ X , its spectrum σ(A) only depends on the algebraic structure of X . If X is aC∗–algebra, then

A = A∗A1/2 = r(A∗A)1/2,

because A∗A is self–adjoint.

Theorem 6.69Let Y ⊂ X be a C∗–subalgebra with identity of a C∗–algebra X . For all B ∈ Y, σ(B) =σX (B) = σY(B).

Remark 6.70Let X be a C∗–algebra with identity. Theorem 1.52 says that the spectrum σ(A) of A ∈ X onlydepends on the smallest C∗–algebra including A.

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6.5.3 C∗–Homomorphism and representation of C∗–algebras:

Definition 6.71 (∗–homomorphism)Let X ,Y be ∗–algebras with identity. A ∗–homomorphism is a map ϕ : X → Y, such that, forall A1, A2 ∈ X and α ∈ C:(i) ϕ(A1 + αA2) = ϕ(A1) + αϕ(A2).(ii) ϕ(A1A2) = ϕ(A1)ϕ(A2).(iii) ϕ(A∗

1) = ϕ(A1)∗.(iv) ϕ(1X ) = 1Y .If it is bijective then it is a ∗–isomorphism and a ∗–isomorphism from X to X is called aautomorphism of the ∗–algebra X .

Remark 6.72Let X ,Y be ∗–algebras with identity and ϕ : X → Y be a ∗–isomorphism. Then ϕ−1 : Y → Xis also a ∗–isomorphism.

Definition 6.73 (Representation of ∗–algebras) Let X be a ∗–algebra with identity.(i) A pair (π,H), where H and π : X → B(H) are a Hilbert space and a ∗–homomorphismrespectively, is called a representation of the ∗–algebra X .(ii) A representation (π,H) is faithful if π is injective.(iii) Let (π,H) be a representation of X . If

π(X )Ω = H

for some Ω ∈ H then Ω is called a cyclic vector and the triplet (π,H,Ω) is called a cyclicrepresentation of X .(iv) The representation (π,H) is irreducible if X = 0 and X = H are the unique subspace ofH such that

π(X )X ⊂ X,

i.e., 0 and H are the unique invariant subspace of the representation (π,H).(v) Two representations (π1,H1) and (π2,H2) of X are equivalent if there is a unitary mapU : H1 → H2 such that π1 = U∗π2U .(vi) Two cyclic representations (π1,H1,Ω1) and (π2,H2,Ω2) of X are equivalent if there is aunitary map U : H1 → H2 such that π1 = U∗π2U and Ω2 = UΩ1.

Remark 6.74(i) If (π,H) is an irreducible representation then all vectors Ω ∈ H\0 are cyclic.(ii) [Gelfand–Naimark’s theorem] Any C∗–algebra has a faithful representation.(iii) C∗–algebras admit, in general, infinitely many non–equivalent irreducible representations.

Theorem 6.75Let X ,Y be C∗–algebras with identity and ϕ : X → Y be a ∗–homomorphism.(i) For all A ∈ X , σ(ϕ(A)) ⊂ σ(A) and ϕ(A)Y ≤ AX . In particular, ϕ is continuous.(ii) If ϕ is injective then σ(ϕ(A)) = σ(A) and ϕ(A)Y = AX .

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Theorem 6.76 (Continuous functional calculous)Let X be a C∗–algebra and A = A∗ ∈ X . Then there is a unique injective ∗–homomorphismϕcont : C(σ(A)) → X such that, for all polynomials P ∈ C(σ(A)), ϕcont(P) = P(A).

Definition 6.77ϕcont is called the continuous functional calculus of the s.a. A = A∗ ∈ X and, for all f ∈C(σ(A)), f(A) := ϕcont(f) ∈ X .

Lemma 6.78Let X be a C∗–algebra and A = A∗ ∈ X . Then

X (A) := f(A) : f ∈ C(σ(A))

is the smallest C∗–subalgebra of X including A.

Theorem 6.79Let X ,Y be C∗–algebra with identity, ϕ : X → Y be a ∗–homomorphism, A = A∗ ∈ X andf ∈ C(σ(A)). Then:(i) σ(f(A)) = f(σ(A)) := f(λ) : λ ∈ σ(A).(ii) ϕ(f(A)) = f(ϕ(A)). Observe that σ(ϕ(A)) ⊂ σ(A) (Theorem 6.75).

Theorem 6.80 (Measurable functional calculous)Let H be an Hilbert space, X ⊂ B(H) be a C∗–algebra and A = A∗ ∈ X . There is a unique (ingeneral, non injective) ∗–homomorphism ϕmess : M(σ(A)) → X , such that:(i) ϕmess|C(σ(A)) = ϕcont.(ii) For all sequences fn ∈ M(σ(A)), n ∈ N, with

supn∈N

fn < ∞

and fn(λ) → f(λ), n → ∞, for all λ ∈ σ(A), we have

ϕmess(fn)(x) → ϕmess(f)(x), n → ∞,

for all x ∈ H.

Definition 6.81ϕmess is called the measurable functional calculus of the s.a. A = A∗ ∈ X and, for all f ∈M(σ(A)), f(A) := ϕmess(f) ∈ X .

6.5.4 The positive elements of a C∗–algebra:

Definition 6.82 (Positive Elements)Let X be a C∗–algebra with identity. The element A ∈ X is positive if A = A∗ and σ(A) ⊂ R+

0

(cf. Lemma 1.14). The set of all positive elements of X is denoted by X+.

84

Theorem 6.83Let X be a C∗–algebra with identity.(i) X+ is a closed set in X with 0 ∈ X+.(ii) X+ is a cone, i.e., for all α ∈ R+

0 and all A1, A2 ∈ X+: αA ∈ X+ and A1 + A2 ∈ X+.

Example 6.84Let K be a compact set and X = C(K). Then, for all f ∈ X , σ(f) = f(K) and

X+ = f ∈ C(K) : f(x) ≥ 0 ∀x ∈ K.

Definition 6.85 (Order relation)Let X be a C∗–algebra with identity and A,B ∈ X . A ≥ B if A− B ∈ X+.

Lemma 6.86Let X be a C∗–algebra with identity. The relation ≥ is a partial order in X , compare withAxiom 6.

Proof: The reflexivity of ≥ follows from the fact that 0 ∈ X+ and the transitivity results fromthe cone properties of X+. Let A,B ∈ X with A ≥ B and B ≥ A. Then,

(A− B) ∈ X+, −(A− B) ∈ X+.

Hence, per Definition, (A− B) = (A− B)∗ and

σ(A− B) ⊂ R+0 , σ(−(A− B)) ⊂ R+

0 .

By Theorem 1.50 (iv), it follows that σ(A − B) = 0, i.e., r(A − B) = 0. By Theorem 1.50(i), the latter yields A− B = 0.

Theorem 6.87 (Characterization of positivity)Let X be a C∗–algebra with identity and A ∈ X . The following propositions are equivalent:(i) A ∈ X+.(ii) There is H ∈ X+ such that A = H2.(iii) There is B ∈ X such that A = B∗B.

Lemma 6.88Let X be a C∗–algebra with identity and A,B ∈ X+ with 0 /∈ σ(A). Then

log(A+B)− log(A) ∈ X+.

Proof: For all c,M > 0,

log(c) = log(M + c)−

M

0

(s+ c)−1ds.

85

From functional calculus,

log(A+B)− log(A)

= limM→∞

M

0

((A+ s)−1 − (A+B + s)−1)ds

+(log(M + A+B)− log(M + A))

= limM→∞

M

0

((A+ s)−1 − (A+B + s)−1)ds

+(log(1 + (A+B)/M)− log(1 + A/M))

= limM→∞

M

0

((A+ s)−1 − (A+B + s)−1)ds

.

Then we obtain that

log(A+B)− log(A)

=

∞

0

(A+ s)−1B(A+B + s)−1ds.

Observe that, for all C,D ∈ X+, σ(CD), σ(DC) ⊂ R+0 : . Indeed, by Lemma 6.66 (iii)

σ(CD) = σ(C1/2C1/2D) ⊂ σ(C1/2DC1/2) ∪ 0.

Since C1/2DC1/2 = (D1/2C1/2)∗(D1/2C1/2) is a positive element, σ(CD) ∈ R+0 . In the same

way, σ(DC) ∈ R+0 . In particular, as

((A+ s)−1B(A+B + s)−1) = (A+ s)−1 − (A+B + s)−1

is self–adjoint, for all s ≥ 0,

(A+ s)−1B(A+B + s)−1 ≥ 0

and thus log(A+B)− log(A) ≥ 0.

6.5.5 Entropy and conditional expectations in C∗–algebras

From H. Araki and H. Moriya Equilibrium Statistical Mechanics of Fermion Lattice Systems.Rev. Math. Phys. 15 (2003) 93–198.

Definition 6.89 (Tracial state)Let X be a C∗–algebra. A state ρ ∈ X ∗

+,1 is tracial if

ρ(AB) = ρ(BA)

for all A,B ∈ X .

86

Lemma 6.90 (Normalized trace)Let X = B(Cn) for n ∈ N. Then,

tr(A) :=1

nTr(A)

defines a tracial state of X , which is faithful.

Proof: See Example 1.63 (ii).

Definition 6.91Let X be a C∗–algebra ∗–isomorph to B(Cn) for n ∈ N, and ϕ : X → B(Cn) be any ∗–isomorphism. The tracial state trX ∈ X ∗

+,1 is the state defined by

trX := tr ϕ.

Remark 6.92Let X be a C∗–algebra ∗–isomorph to B(Cn) for n ∈ N, and ϕ,ϕ : X → B(Cn) be two arbitrary∗–isomorphism. One can show (see Lemma 2.9) that there is a unitary matrix U such that

ϕ(A) = U∗ϕ(A)U

for all A ∈ X . Then,tr(ϕ(A)) = tr(U∗ϕ(A)U) = tr(ϕ(A))

and the definition of trX does not depend on the choice of ϕ.

Theorem 6.93 (tracial states for US/F )The C∗–algebras US/F has at least one faithful tracial state tr = trUS/F ∈ (US/F )∗+,1.

Idea of the proof: For all Λ ⊂ Λ ∈ Pf , per definition we have

trUS/F

Λ

|US/FΛ

= trUS/FΛ

.

Then we define a normalized linear functional trUS/F : US/F

0 → C by

trUS/F (A) := lim→∞

trUS/FΛ

(A).

Its (unique) extension trUS/F : US/F → C has the desired properties of a tracial state.

Remark 6.94(i) trUS/F has the following property: for all disjoint sets Λ1,Λ2 ∈ Pf and A ∈ US/F

Λ1, B ∈ US/F

Λ2,

trUS/F (AB) = trUS/F (A)trUS/F (B).

(ii) trUF is an even state.

87

Lemma 6.95Let X be a C∗–algebra isomorph to B(Cn) for n ∈ N. Then, for any state ρ ∈ X ∗

+,1, there is a

unique positive Dρ ∈ X+ such that

ρ(A) = trX (DρA), ∀A ∈ X .

ρ ∈ X ∗+,1 is called regular when 0 /∈ σ(Dρ).

Proof: As for Lemma 2.3.

Definition 6.96 (Normalized and relative entropy)Let X be a C∗–algebra isomorph to B(Cn) for n ∈ N. Then:(i) For all ρ ∈ X ∗

+,1 the normalized entropy is

S(ρ) = −ρ(log(Dρ)) := trX (η(Dρ)).

[η(x) := −x log(x).](ii) For ρ ∈ X ∗

+,1 and all regular states ρ ∈ X ∗+,1, the relative entropy of ρ with respect to ρ is

defined by

S(ρ, ρ) = ρ(log(Dρ)− log(Dρ))

:= −trX (η(Dρ))− ρ(log(Dρ)).

If ρ is not regular, S(ρ, ρ) can be defined via some limit.

Remark 6.97(i) S(ρ) = S(ρ)− log(n) = S(ρ)− S(trX ) = −S(trX , ρ).(ii) The relative entropy is non–negative and S(ρ, ρ) = 0 when ρ = ρ.

Definition 6.98 (Schwarz–maps)Let X be a C∗–algebra with identity and Y ⊂ X be a C∗–subalgebra with identity. A linear mapE : X → Y is a Schwarz–map when E(1) = 1 and

E(A∗A) ≥ E(A)∗E(A) ≥ 0

for all A ∈ X .

Theorem 6.99 (Schwarz–maps and entropy)Let X be a C∗–algebra isomorph to B(Cn) for n ∈ N and Y ⊂ X be a C∗–subalgebra isomorph toB(Cm) for m ≤ n. Let E : X → Y be a Schwarz–map and ρ, ρ ∈ X ∗

+,1. Then ρE, ρ E ∈ X ∗+,1

andS(ρ E, ρ E) ≤ S(ρ, ρ).

Proof: ρ E, ρ E ∈ X ∗+,1 is let as an exercise. For the inequality of the theorem, see Theorem

1.5 in: M. Ohya, D. Petz. Quantum Entropy and its Use. Springer. 1993.

88

Definition 6.100 (Conditional expectations)Let X be a C∗–algebra with identity and Y ⊂ X be a C∗–subalgebra with identity. Let tr be afaithful tracial state of X . A conditional expectation w.r.t. tr is the map EX

Y : X → Y satisfying

tr(AB) = tr(EXY (A)B), (6.22)

for all A ∈ X and B ∈ Y.

Theorem 6.101Let X be a C∗–algebra with identity and Y ⊂ X be a C∗–subalgebra with identity. Let tr be afaithful tracial state of X . A conditional expectation EX

Y : X → Y has the following properties(i) EX

Y is linear and unique.(ii) EX

Y (AB) = EXY (A)B and EX

Y (BA) = BEXY (A) for all A ∈ X and B ∈ Y.

(iii) EXY (X+) ⊂ Y+, EX

Y (1) = 1, and EXY (A

∗) = EXY (A)

∗ for all A ∈ X .(iv) EX

Y EXY = EX

Y and EXY = 1.

(v) EXY is a Schwarz–map, i.e., for all A ∈ X , EX

Y (A∗A) ≥ EX

Y (A)∗EX

Y (A).

Proof: Uniqueness: Let A ∈ X , A, A ∈ Y , such that

tr(AB) = tr(AB) = tr(AB)

for all B ∈ Y . Thentr(B(A − A)) = 0

for all B ∈ Y . In particular,tr((A − A∗(A − A)) = 0.

As tr is faithful, one gets A − A = 0.

Linearity: Let A = αA1 + βA2, α, β ∈ C, A1, A2 ∈ X . Then, for all B ∈ Y ,

tr(AB) = tr(EXY (A)B)

= αtr(A1B) + βtr(A2B)

= αtr(EXY (A1)B) + βtr(EX

Y (A2)B)

= tr(αEX

Y (A1) + βEXY (A2))B

.

Using the uniqueness of EXY we arrive at the equality

EXY (A) = αEX

Y (A1) + βEXY (A2).

(ii): Now, for all A ∈ X and B,B ∈ Y ,

tr(ABB) = tr(EXY (AB)B) = tr(EX

Y (A)BB).

Then, for A ∈ X and B ∈ Y ,EX

Y (AB) = EXY (A)B.

89

On the other hand, for all A ∈ X and B,B ∈ Y ,

tr(BAB) = tr(EXY (BA)B)

= tr(ABB)

= tr(EXY (A)B

B)

= tr(BEXY (A)B

).

It follows thatEX

Y (BA) = BEXY (A).

(iii)–(iv): For A ∈ Y ⊂ X ,

tr(EXY (A)B) = tr(AB), ∀B ∈ Y .

Using the uniqueness of EXY we obtain EX

Y (A) = A for all A ∈ Y ⊂ X . In particular, EXY :

X → Y is surjective, EXY (1) = 1 and EX

Y EXY = EX

Y .

Positivity: Let (π,H,Ω) be a GNS–representation of tr|Y . For all A ∈ X+ and B ∈ Y ,

π(B)Ω, π(EAB (A))π(B)Ω

= tr(B∗EXY (A)B)

= tr(EXY (A)BB∗)

= tr(ABB∗)

= tr(B∗AB) ≥ 0.

Since π(Y)Ω is dense in H, we deduce π(EXY (A)) ≥ 0. Since tr is faithful, the GNS–

representation is also faithful and EXY (A) is positive.

Now, using again that GNS–representation is faithful, for all A ∈ X ,

EAB (A)= π(EX

Y (A))= sup

B,B∈Y

|π(B)Ω, π(EX

Y (A))π(B)Ω| :

π(B)Ω, π(B)Ω ≤ 1

= supB,B∈Y

|tr(B∗EX

Y (A)B)| :

tr(B∗B), tr((B∗B) ≤ 1

= supB,B∈Y

|tr(B∗AB)| :

tr(B∗B), tr((B∗B) ≤ 1

= A.

Observe that one need the GNS–representation of tr (instead of tr|Y) to show the last equation.90

For all A ∈ X and B ∈ Y ,

tr(EXY (A

∗)B) = tr(A∗B)

= tr(B∗A)

= tr(AB∗)

= tr(EXY (A)B

∗)

= tr(BEXY (A)

∗)

= tr(EXY (A)

∗B).

Using the uniqueness of EXY (A

∗) we obtain

EXY (A

∗) = EXY (A)

∗.

(v): Using the positivity of EXY , for all A ∈ X ,

0 ≤ EXY ((A− EX

Y (A))∗(A− EX

Y (A)))

= EXY

A∗A− A∗EX

Y (A)

−EXY (A)

∗A+ EXY (A)

∗EXY (A)

= EXY (A

∗A)− EXY (A

∗)EXY (A)

−EXY (A)

∗EXY (A) + EX

Y (A)∗EX

Y (A)

= EXY (A

∗A)− EXY (A)

∗EXY (A).

In other words,EX

Y (A∗A) ≥ EX

Y (A)∗EX

Y (A).

Theorem 6.102For all Λ ∈ Pf there is a unique conditional expectation EΛ : U → UΛ (with U = US/F ) w.r.t.tr. Moreover, for all Λ,Λ ∈ Pf ,

EΛEΛ = EΛEΛ = EΛ∩Λ .

Proof: Let U = US. For any (not necessarily finite) subset Γ ⊂ L we define the C∗–subalgebra

UΓ :=

Λ∈Pf , Λ⊂Γ

UΛ ⊂ U .

Let Λ ∈ Pf and AnNn=1 be a basis of the finite dimensional space UΛ. For all A ∈ U , thereare unique elements A

n∈ UL\Λ, n = 1, . . . , N such that

A =N

n=1

AnAn.

91

Define the linear map EΛ : U → UΛ by

EΛ(A) = EΛ(N

n=1

AnAn) :=

N

n=1

tr(An)An.

For all A ∈ X and B ∈ UΛ (because of [B,An] = 0 and the product structure of tr (Remark

6.94 (i)), we have

tr(AB) = tr

N

n=1

(AnB)An

=N

n=1

tr(AnB)tr(An)

= tr

N

n=1

tr(An)An

B

= tr(EΛ(A)B).

By Theorem 6.101, EΛ is the unique conditional expectation U → UΛ w.r.t. tr. In particular,EΛ does not depend on the basis AnNn=1 of UΛ.

For all Λ,Λ ∈ Pf , one has

EΛ(A) ∈ UΛ∩Λ , ∀A ∈ UΛ .

SinceUΛ = UΛ∩Λ ⊗ UΛ\Λ,

it remains to prove (because of linearity) A ∈ UΛ has the form A = A1A2 with A1 ∈ UΛ∩Λ,A2 ∈ UΛ\Λ ⊂ UL\Λ. By Theorem 6.101 (ii)

EΛ(A1A2) = A1EΛ(A2)

= A1EΛ(1A2)

= A11tr(A2) ∈ UΛ∩Λ.

For all A ∈ U , one has EΛ(A) ∈ UΛ and thus,

EΛ(EΛ(A)) ∈ UΛ∩Λ .

Then, for all B ∈ UΛ∩Λ ,

tr(EΛ(EΛ(A))B) = tr(EΛ(A)B)

= tr(AB)

= tr(EΛ∩Λ(A)B).

Using the uniqueness of the conditional expectation (Theorem 6.101 (i)) we obtain that

EΛ(EΛ(A)) = EΛ∩Λ(A).

The proof when U = UF is similar. The even and odd element in the construction of EΛ

must be analyzed separately, see proof of Theorem 4.7 in Araki, Moriya 2003 for more details.

92

Corollary 6.103For any Λ1,Λ2 ∈ Pf , there is a conditional expectation EΛ : US/F

Λ1∪Λ2→ US/F

Λ , Λ ⊂ Λ1∪Λ2, suchthat:(i) EΛ1 EΛ2 = EΛ2 EΛ1 = EΛ1∩Λ2, EΛ1∪Λ2 = 1.(ii) S(ρΛ)− S(ρΛ) = S(ρΛ EΛ, ρΛ EΛ) for all

Λ ⊂ Λ ⊂ Λ ⊂ Λ1 ∪ Λ2.

[cf. proof of Theorem 6.105.]

Proof: We identify the conditional expectation EΛ of Theorem 6.102 with its restrictionEΛ|UΛ1∪Λ2

. It is a conditional expectation UΛ1∪Λ2 → UΛ for all Λ ⊂ Λ1 ∪ Λ2. (i) is provenin Theorem 6.102.

For allΛ ⊂ Λ ⊂ Λ1 ∪ Λ2

and ρ ∈ (UΛ1∪Λ2)∗+,1,

ρ|UΛ EΛ = ρ|Λ ⊗ trUΛ1∪Λ2\Λ

.

SinceUΛ1∪Λ2 = UΛ ⊗ UΛ1∪Λ2\Λ,

it remains to prove (because of linearity) A ∈ UΛ1∪Λ2 has the form A = A1A2 with A = A1A2,A1 ∈ UΛ, A2 ∈ UΛ1∪Λ2\Λ. By Theorem 6.101 (ii),

ρ|UΛ(EΛ(A1A2)) = ρ(EΛ(A1A2))

= ρ(A1EΛ(A2))

= ρ(A1EΛ(1A2))

= ρ(A11tr(A2))

= ρ(A1)tr(A2)

= ρ|Λ(A1)trUΛ1∪Λ2\Λ(A2)

= ρ|UΛ ⊗ trUΛ1∪Λ2\Λ(A1A2)

Observe that the 4th equation is proven in Theorem 6.102.

Let ρ ∈ (UΛ1∪Λ2)∗+,1 and

Λ ⊂ Λ ⊂ Λ ⊂ Λ1 ∪ Λ2.

Observe thatDρ|UΛ

⊗trUΛ1∪Λ2\Λ= Dρ|UΛ

⊗ 1UΛ1∪Λ2\Λ

andlog(Dρ|UΛ

⊗ 1UΛ1∪Λ2\Λ) = log(Dρ|UΛ

)⊗ 1UΛ1∪Λ2\Λ.

The same holds for the set Λ. Then, per definition,

S(ρ|UΛ ⊗ trUΛ1∪Λ2\Λ, ρ|UΛ ⊗ trUΛ1∪Λ2\Λ

)

= ρ(log(Dρ|UΛ)⊗ 1UΛ1∪Λ2\Λ

− log(Dρ|UΛ

)⊗ 1UΛ1∪Λ2\Λ)

= −ρ|UΛ(log(Dρ|UΛ)) + ρ|UΛ

(log(Dρ|UΛ))

= S(ρ|UΛ)− S(ρ|UΛ).

93

Therefore

S(ρ|UΛ EΛ, ρ|UΛ

EΛ)

= S(ρ|UΛ EΛ)− S(ρ|UΛ

EΛ).

Theorem 6.104 (Quasi–convexity of S)Let ρ, ρ ∈ X ∗

+,1 with X isomorph to B(Cn) for n ∈ N. Then, for all λ ∈ [0, 1],

S(λρ+ (1− λ)ρ) ≤ λS(ρ) + (1− λ)S(ρ) + η(λ) + η(1− λ).

Proof: We assume that ρ and ρ are regular. The general case follows via some limit. ByLemma 6.88, we have, for all λ ∈ (0, 1),

log(λDρ + (1− λ)Dρ) ≥ log(λDρ).

In particular,

−λTr(Dρ log(λDρ + (1− λ)Dρ))

≤ −λTr(Dρ log(λDρ))

= −λ log(λ) + λS(ρ).

On the other hand,

−(1− λ)Tr(Dρ log(λDρ + (1− λ)Dρ))

≤ −(1− λ)Tr(Dρ log((1− λ)Dρ))

= −(1− λ) log(1− λ) + (1− λ)S(ρ).

So, the theorem follows from the addition of the last two upper bounds.

Theorem 6.105 (Strong subadditivity of S)For all Λ1,Λ2 ∈ Pf and all states ρΛ1∪Λ2

∈ (US/F

Λ1∪Λ2)∗+,1,

S(ρΛ1∪Λ2)− S(ρΛ1

)− S(ρΛ2) + S(ρΛ1∩Λ2

) ≤ 0.

Here, for all Λ ⊂ Λ1 ∪ Λ1, ρΛ := ρΛ1∪Λ2|US/F

Λ.

Proof: By Theorem 6.101, there is a Schwarz–map

EΛ : US/F

Λ1∪Λ2→ US/F

Λ

with Λ ⊂ Λ1 ∪ Λ2 such that

EΛ1 EΛ2 = EΛ2 EΛ1 = EΛ1∩Λ2 , EΛ1∪Λ2 = id

94

andS(ρΛ)− S(ρΛ) = S(ρΛ EΛ, ρΛ EΛ)

for all Λ ⊂ Λ ⊂ Λ ⊂ Λ1 ∪ Λ2. It follows by Theorem 6.99 that

S(ρΛ2)− S(ρΛ1∪Λ2

)

= S(ρΛ1∪Λ2 EΛ2 , ρΛ1∪Λ2

EΛ1∪Λ2)

= S(ρΛ1∪Λ2 EΛ2 , ρΛ1∪Λ2

)

≥ S(ρΛ1∪Λ2 EΛ2 EΛ1 , ρΛ1∪Λ2

EΛ1)

= S(ρΛ1∪Λ2 EΛ1∩Λ2 , ρΛ1∪Λ2

EΛ1)

= S(ρΛ1∩Λ2)− S(ρΛ1

).

Corollary 6.106 (Subadditivity of S)For all Λ1,Λ2 ∈ Pf an all ρΛ1∪Λ2

∈ (US/F

Λ1∪Λ2)∗+,1,

S(ρΛ1∪Λ2) ≤ S(ρΛ1

) + S(ρΛ2).

Proof: For all Λ ∈ Pf :

US/F

Λ1∪Λ2∼ B(CnΛ)

with nΛ = 2|S||Λ|. As the von Neumann entropy is non–negative, it follows from Theorem 6.105and Remark 6.97 (i) that

S(ρΛ1∪Λ2) ≤ S(ρΛ1

) + S(ρΛ2) + |S||Λ1 ∩ Λ2| log(2).

Then, by Remark 6.97 (i),

S(ρΛ1∪Λ2)− |S|(|Λ1 ∪ Λ2|+ |Λ1 ∩ Λ2|) log(2)

≤ S(ρΛ1)− |S||Λ1| log(2) + S(ρΛ2

)− |S||Λ2| log(2).

In particular,

S(ρΛ1∪Λ2)− |S|(|Λ1|+ |Λ2|) log(2)

≤ S(ρΛ1)− |S||Λ1| log(2) + S(ρΛ2

)− |S||Λ2| log(2).

Remark 6.107For a product state, the von Neumann Entropy is additive, i.e., for all disjoint sets Λ1,Λ2 ∈ Pf

and ρΛi∈ (US/F

Λi)∗+,1, i = 1, 2, we have that

S(ρΛ1⊗ ρΛ2

) = S(ρΛ1) + S(ρΛ2

).

95

Corollary 6.108 (Concavity of the entropy)For all Λ ∈ P, the map ρ → S(ρ) from (US/F

Λ )∗+,1 to R+0 is concave, i.e., for all ρ, ρ ∈ (US/F

Λ )∗+,1

and all λ ∈ [0, 1],S(λρ+ (1− λ)ρ) ≥ λS(ρ) + (1− λ)S(ρ).

Proof: Let ω = λρ + (1 − λ)ρ. There is a set Λ ∈ Pf disjoint from Λ (Λ ∩ Λ = ∅) and

ω ∈ (US/F

Λ∪Λ)∗+,1 such thatω|US/F

Λ= ω

andS(ω)− S(ω|US/F

Λ

) = λS(ρ) + (1− λ)S(ρ).

This can be seen as follows: Let any set Λ ∈ Pf such that Λ ∩ Λ = ∅ and |S||Λ| ≥ 2. Choose

e, e ∈ (US/F

Λ )+ such that ee = ee = 0, ee = e, ee = e and tr(e) = tr(e−|S||Λ|). Define

D := λDρ(2|S||Λ|e) + (1− λ)Dρ(2

|S||Λ|e).

Because of the product structure of tr (see Remark 6.94 (i)) the equality ω(A) := tr(DA) definesa state of US/F

Λ∪Λ such that ω|US/FΛ

= ω. Observe that, in any representation of US/F

Λ ∼ B(CnΛ ),

e and e are two different orthogonal projections of dimension 1. In any representation of thetype

US/F

Λ∪Λ ∼ B(CnΛ)⊗ B(CnΛ ),

we haveD = λDρ ⊗ e+ (1− λ)Dρ ⊗ e.

It follows from the definition of the Von–Neumann entropy that

S(ω)− S(ω|US/F

Λ

) = λS(ρ) + (1− λ)S(ρ). (6.23)

Using (6.23) and the subadditivity of the entropy we obtain

(S(ω) + S(ω|US/FΛ

))− S(ω|US/FΛ

)

≥ λS(ρ) + (1− λ)S(ρ).

6.6 Additional details on Gibbs state

6.6.1 Gibbs state as KMS (Kubo-Martin-Schwinger) states

Definition 6.109Let S be a finite system with OS = OX and Hamiltonian H ∈ OX . For any t ∈ R the∗–automorphisms αH

tof X are defined by

αH

t(A) := exp(itH)A exp(−itH).

The pair (X , αH

tt∈R) creates a C∗–dynamical system.

96

Remark 6.110Let S be a finite system with OS = OX and Hamiltonian H ∈ OX .(i) The Gibbs state ωH,β is stationary w.r.t. αH

tt∈R, i.e., for all t ∈ R, ωH,β αH

t= ωH,β

because

ωH,β αH

t(A) = Tr(DH,β exp(itH)A exp(−itH))

= Tr(exp(itH)DH,βA exp(−itH))

= Tr(exp(−itH) exp(itH)DH,βA)

= Tr(DH,βA).

(ii) For any A,B ∈ X , the function GA,B : R → C defined by

GA,B(t) := ωH,β(AαH

t(B)).

can be uniquely extended to an analytic function C → C. This follows by series expansion ofGA,B(t).

Theorem 6.111 (KMS property of ωH,β)Let S be a finite system with OS = OX and Hamiltonian H ∈ OX . Then, for any A,B ∈ X ,there is a continuous function

GA,B : R+ i[0, β] → C

such that:(i) For all t ∈ R,

GA,B(t) = ωH,β(AαH

t(B)),

GA,B(t+ iβ) = ωH,β(αH

t(B)A),

(ii) GAB is analytic on R+ i[0, β] ⊂ C.

GA,B(t) = ωH,β(AαH

t(B)),

GA,B(t+ iβ) = ωH,β(αH

t(B)A),

Proof: Define

GA,B(z) := ωH,β(AαH

z(B))

= Tr(DH,βA exp(izH)B exp(−izH))

= Z−1H,β

Tr(exp(−βH)A exp(izH)B exp(−izH)).

By Remark 6.110, the assertion (ii) holds and per Definition

GA,B(t) = ωH,β(AαH

t(B))

97

for all t ∈ R. Then

GA,B(t+ iβ)

= Z−1H,β

Tr(exp(−βH)A exp(i(t+ iβ)H)B exp(−i(t+ iβ)H))

= Z−1H,β

Tr(B exp(−i(t+ iβ)H) exp(−βH)A exp(i(t+ iβ)H))

= Z−1H,β

TrB exp(−itH) exp(βH) exp(−βH)

A exp(itH) exp(−βH)

= Z−1H,β

Tr(exp(−βH)B exp(−itH)A exp(itH))

= ωH,β(BαH

−t(A)).

As ωH,β is stationary, we have that

ωH,β(BαH

−t(A)) = ωH,β(α

H

t(BαH

−t(A))) = ωH,β(α

H

t(B)A)

and (i) follows.

Remark 6.112Conditions (i) and (ii) are called the (β,αH)–KMS conditions of the state ωH,β.

6.6.2 Gibbs state and zero–law of thermodynamics

Remark 6.113 (zero–law of thermodynamics)The zero–law of thermodynamics postulates the existence of thermal reservoirs: For any tem-perature β−1 > 0 there is a system ST with the property that, for any finite system S afterturning adiabatically on and off the contact between S and ST , the system S will be in theequilibrium state corresponding to the temperature β−1.

Lemma 6.114 (Product states)Let X1 ⊂ B(H1) and X2 ⊂ B(H2) be two C∗–algebra with identity and ω1 ∈ X ∗

1 , ω2 ∈ X ∗2 two

states. Assume that dim(Hi) < ∞ for one i = 1, 2. Then there is a unique state ω1 ⊗ ω2 onX1 ⊗ X2 ⊂ B(H1 ⊗H2) such that

(ω1 ⊗ ω2) (A1 ⊗ A2) = ω1(A1)ω2(A2) (6.24)

for all A1 ∈ X1, A2 ∈ X2.

Proof: From the density of the set

linA1 ⊗ A2 : A1 ∈ X1, A2 ∈ X2 ⊂ X1 ⊗ X2,

and the continuity and linearity of states, if the state ω1 ⊗ ω2 exists, it must be unique.

Observe that there is a unique ∗–Homomorphism

π1 ⊗ π2 : X1 ⊗ X2 → B(H1)⊗ B(H2)98

such thatπ1 ⊗ π2(A1 ⊗ A2) = π1(A1)⊗ π2(A2).

See Lemma 1.100. Here, for i = 1, 2, (πi, Hi,Ωi) is the GNS representation of ωi.

Define the linear functional ω : X1 ⊗ X2 → C by

ω(C) := Ω1 ⊗ Ω2, π1 ⊗ π2(C)Ω1 ⊗ Ω2.

As Ω1 ⊗ Ω2 = 1 and π1 ⊗ π2 is a ∗–Homomorphism, ω is a state. Per construction ω is astate satisfying (6.24):

ω(A1 ⊗ A2) = Ω1 ⊗ Ω2, π1(A1)⊗ π2(A2)Ω1 ⊗ Ω2= Ω1, π1(A1)Ω1Ω2, π2(A2)Ω2.

Lemma 6.115 (Partial trace)Let X ⊂ B(H1) and Y ⊂ B(H2) be two C∗–algebra with identity and let ω ∈ (X ⊗ Y)∗ be astate. Then the linear functional ωX : X → C defined by

ωX (A) := ω(A⊗ 1Y)

is a state of X .

Proof: The linearity of ωX is clear. Moreover, as

ωX (1X ) = ω(1X ⊗ 1Y) = 1,

andωX (A

∗A) = ω((A⊗ 1Y)∗(A⊗ 1Y)) ≥ 0

ωX is a positive and normalized linear functional.

Theorem 6.116 (Existence of reservoirs)For all β ∈ (0,∞), there is a C∗–dynamical system (X ,αt) and a state ωβ of X such that, forall n ∈ N, H ∈ OY with Y = B(Cn) and all states ω0 on Y,

limt→∞

ωt = ωH,β.

Hereωt := (ωt)Y , ωt := (ωβ ⊗ ω0) αW

t,0.

(X ⊗ Y ,αW

t,s) is the non–autonomous C∗–dynamical system defined by

d

dtαW

t,s(A) = L(t)αW

t,s(A).

withL(t) = L0 +W (t),

being the sum of the generator L0 of the autonomous dynamical system (X ⊗ Y ,αt ⊗ αH

t) and

the time–dependant generator W (t) in X ⊗ Y which satisfies W (t) = 0 for all t ∈ R as well as

limt→∞

W (t) = 0.

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The term W (t) represent the coupling between systems X and Y which can be chosen in avery general class. See Paragraph 1.4.

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