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Notes on Operator Algebras

G. Jungman

School of Natural Sciences, Institute for Advanced Study

Olden Lane, Princeton, NJ 08540

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Contents

1 Structure Theory I 4

1.1 Invertible Elements and Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Gelfand Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Local Algebras, Idempotents and Projections . . . . . . . . . . . . . . . . . . . . . 9

2 von Neumann Algebras 11

2.1 Commutant and Bicommutant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Hyperfinite Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 States and Representations 14

3.1 GNS Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Basic Structure of Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Structure Theory II 21

4.1 Weights and Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.2 Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Matrices 27

5.1 Inductive Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.2 Glimm Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5.3 Matn (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6 Automorphism Groups 31

6.1 Automorphisms and Invariant States . . . . . . . . . . . . . . . . . . . . . . . . . . 31

6.2 KMS States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.3 Modular Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6.4 Type III Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2

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CONTENTS 3

6.5 Hyperfiniteness Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

7 Extensions 49

8 K-Theory 508.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.2 Commutative K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.3 K0-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.4 K1-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

8.5 AF Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.6 Equivariant K-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

8.7 Index Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

9 Nuclear C Algebras 59

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Chapter 1

Structure Theory I

1.1 Invertible Elements and Spectra

Definition. A C algebra is an involutive Banach algebra A with a norm satisfying the relations

AB A B ,

A = A ,

AA = A2 .

We denote the dual algebra by A. Then we can and often will use the weak topology for A andthe weak-* topology for A. These properties are motivated by the study of bounded operators onHilbert spaces. For example, to show that AA = A2 holds in B(H), for some Hilbert space H,

we have the following calculation.

A2 2 AA

= A2 AA .

The first two properties are easy and they give

AA A A = A2

= A2 = AA .

Remark. Some algebras come without a unit element. Such algebras are sometimes more easy todeal with if a unit element is appended in a formal way. A typical example of this is the appending

of the Dirac delta function to a convolution algebra of smooth functions, there being no smooth

representative which can play the role of the unit element in a convolution algebra.

Definition. Let A be a Banach algebra. Define the unitization of A to be

A+ = (, ) A C

with

(a, ) (b, ) = (ab + b + a, ).

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CHAPTER 1. STRUCTURE THEORY I 5

Definition. The resolvent ofA A, Res(A) C, is the set { C : ( A)1existsinA}.

Definition. The spectrum ofA A, Spec(A), is the complement ofRes(A).

1.1. Theorem (Spectral Radius).

1. Res(A) is open.

2. Spec(A) is compact.

3. supSpec(A) || = limn An1/n = infn An

1/n.

Proof. The first two follow easily from the definitions. To prove the third, write n = An, thenn+l nl. Let m Z, m > 0. We can write n = p(n)m + q(n), where p(n) and q(n) < m areunique integers. Then

1/nn 1/np(n)m

1/nq(n)

p(n)/nm

1/nq(n).

Now

lim supn

1/nn limn

p(n)/nm (sup{0, 1, . . . , m})1/n lim

np(n)/nm

1/mm , for any m.

Therefore

lim supn

1/nn infn

1/nn lim infn

1/nn ,

so that the limit of the theorem exists and equals infn An1/n

. To show that supSpec(A) || =

limn An

1/n

, let be such that || > limn An

1/n

, then || > infn An

1/n

so that the series(A/)n converges in norm to (1 A/)1 so Spec(A). Furthermore, suppose that for all r

in the interval

supSpec(A) ||, supn An1/n

, (1 A/r)1 existed. Then (1 A/r)1 would be

analytic for r > supSpec(A) || and (1 A/r)1 =

n(A/r)

n. But An/rn1/n > An1/n

infnAn1/n 1.

.

1.2. Theorem (Holomorphic Symbolic Calculus). Let A be a C algebra, and A A. Letf beholomorphic on O Spec(A). Then we can define f(A) A such that

1. f f(A) is a homomorphism of the algebra of holomorphic functions on O to A.

2. For the function f() , f(A) = A.

3. Forf() ( 0)1, f(A) = (A )1.

4. Spec(f(A)) = f(Spec (A)). [spectral mapping theorem]

Proof. First we prove that 12i

( )n( A)1d = ( A)n, where is a contour surroundingSpec(A). Let yn equal this integral expression. Then it is easy to show that ( A)yn = yn+1. Thusthe formula will follow from the case n = 0.

By the Cauchy theorem for vector-valued integrals we can replace the contour by a circle of radiusr > A. Then integrate ( A)1 =n1An term by term. This proves the result for n = 0.

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Now we can assert the statements of the theorem for rational functions f since the integrands willthen always be of the given form by factorization.

Finally, the approximation of holomorphic functions by rational functions converges uniformly on

compact sets. Therefore we can define f(A) for holomorphic f by interchanging the limit and the

integral.Remark. In order to state a more refined version of the symbolic calculus, valid for arbitrary contin-

uous functions on Spec(A), we need to delve into the theory of commutative C algebras.

Definition. A subspaceI of a commutative algebra A is called an ideal if, for any A A, AB Iwhenever B I.

Definition. A complex homomorphism, , of a Banach algebra is a linear functional with the prop-erty (AB) = (A)(B). It is also called a character of the algebra.

1.3. Theorem (Gleason, Kahane, Zelazko). If is a linear functional on a Banach algebra such

that(1) = 1, and(A) = 0 for any invertible A, then is a complex homomorphism.

Proof. See [Rud91].

1.4. Theorem (Gelfand-Mazur). Let A be a Banach algebra in which every nonzero element isinvertible. Then A is isometrically isomorphic to C.

Proof. Let A A and 1 = 2; then at least one of 1 A, 2 A is invertible by hypothesis.Spec(A) is nonempty by a standard result (a spectrum is never empty), so it follows that for eachsuch A there is a unique (A) C in Spec(A). The mapping A (A) is an isomorphism sinceA = (A) 1; it is obviously an isometry.

1.5. Theorem. LetA be a commutative Banach algebra, and let be the set of all complex homo-morphisms ofA. Then

1. Every maximal ideal ofA is the kernel of some h .

2. Ifh , ker(h) is a maximal ideal ofA.

3. A A invertible if and only ifh(A) = 0 for all h .

4. A A invertible if and only ifA lies in no proper ideal of A.

5. Spec(A) if and only ifh(A) = for some h .

Proof.

1. Let M be a maximal ideal ofA. Since the set of all invertible elements is open, maximal idealsare closed; so M is closed. Therefore A/M is a Banach algebra. Choose x A, x M, andset J = {ax + y : a A, y M}. Then J is an ideal, and x J so J is larger than M, soJ = A. Therefore, for some A A and y M we have Ax + y = 1. Applying the quotientmap : A A/M we see that (A)(x) = (1), thus every nonzero element of A/M isinvertible. So by the Gelfand-Mazur theorem A/M = C, j : A/M C. Let h = j , thenh and h(M) = 0.

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2. Ifh then h1(0) is an ideal in A with codimension 1. Therefore it is maximal.

3. IfA is invertible in A and h , then h(A)h(A1) = h(1) = 1 and so h(A) = 0. IfA is notinvertible then {aA : a A} {1} = , so {aA : a A} is a proper ideal which lies in amaximal ideal and so it is annihilated by some h by the first result.

4. No invertible element lies in any proper ideal. The converse is proved in the previous item.

5. Apply the third item to A instead ofA.

Remark. As an application of the above we have the following result on Fourier series.

1.6. Theorem (Wiener Lemma). Suppose f : Rn C,

f(x) = mZn am exp im x, |am| < ,Iff(x) is never zero then

1/f(x) =mZn

bm exp im x,

|bm| < .

Proof. Let A be the commutative Banach algebra of functions of the form

am exp im x with thenorm f =

|am|. For each x Rn, the evaluation map f f(x) is a complex homomorphism.

By assumption no evaluation gives zero. So if we can prove that all complex homomorphisms of Aare evaluations for some x Rn, then the third part of the structure theorem above will assert the

existence of1/f in A.

Let h be any complex homomorphism of A. Write gr(x) = exp ixr, r = 1, . . . , n; xr is the r-thcoordinate ofx Rn. Then gr A, 1/gr A, and gr = 1/gr = 1.

It is easy to see that if A < 1 then |(A)| < 1 for any complex homomorphism , since for any C with || > 1 we know (1 A/)1 exists and so (1 A/) = 0 and so (A) = . So wesee that |h(gr)| 1 and |h(1/gr)| = |1/h(gr)| 1. Therefore h(gr) = exp iyr for some yr R,r = 1, . . . , n.

Let P be any trigonometric polynomial. Then h(P) = P(y1, . . . , yn). But h is continuous and thetrigonometric polynomials are dense in A, so h(f) = f(y) for all f A and so h is evaluation at

y.

1.2 Gelfand Transform

Definition. Given A A we can define a function A : C byA(h) = h(A).

A is called the Gelfand transform of A. It is also sometimes called the spectrum ofA, though wewill never use this terminology.

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Definition. The Gelfand topology on is the weakest topology such that A is continuous for everyA A.

Definition. The radical ofA, rad(A), is the intersection of all the maximal ideals of A.

1.7. Theorem. Let be equipped with the Gelfand topology. Then

1. is a compact Hausdorff space.

2. The Gelfand transform is a homomorphism ofA onto a subalgebra of the continuous functionson , and the kernel of this homomorphism is rad(A). Thus the Gelfand transform is anisomorphism if and only ifrad(A) = {0}.

3. For all A A, Ran(A) = Spec (A), andA

= supSpec(A) || A. Furthermore,

A rad(A) if and only ifsupSpec(A) || = 0.

Proof. For a complete proof see [Rud91]. The second and third items follow from the structuretheorem above, together with some computation. The first item follows from the Banach-Alaoglu

theorem and a proof of the closure of . The Gelfand topology is the restriction of the weak-*topology to .

Definition. The set A is called the spectrum of A. To avoid technical complications, thespectrum ofA is actually defined as not to contain the zero homomorphism.

Remark. Here is an example that shows how the Gelfand transform is a generalization of the Fourier

transform, in the L1 context. Let A = L1 (Rn) dx, with unit attached. So members ofA are f + ,where is the Dirac measure. Of course, the multiplication is convolution.

Let h be a complex homomorphism, h ; then h is one of the following forms,

ht(f + ) = f(t) + or

h(f + ) =

We prove this as follows. If h(f) = 0 for all f A then h = h. Assume h(f) = 0 for some

f A. Then h(f) = f for some L (Rn) dx. Since h(f g) = h(f)h(g), we can show that coincides almost everywhere with a continuous function b which satisfies b(x + y) = b(x)b(y).But every bounded solution of this functional equation is of the form b(x) = exp(ixt). Thus

h(f) = f(t) and h is of the form ht.So = Rn {}, say with the topology of the one-point compactification. Since f(t) 0 as|t| , A C(). A separates points on so the weak topology induced on by A is as wehave chosen.

1.8. Theorem (Gelfand-Naimark). Let A be a commutative C algebra, and equip with theGelfand topology as usual. Then the Gelfand transform is an isometric *-isomorphism of A onto thealgebra of continuous C-valued functions on , C().

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Proof. In order to show that the involution is preserved we need only show that for A = A, h(A) R. So let A = A and write h(A) = + i. Calculate

h(A + it) = + i(+ t)

2

+ (+ t)2

= |h(A + it)|2

A + it

2

= (A + it)(A it) A

2

+ t2

2 + 2 + 2t A2 t R

= = 0.

By definition the elements of A C() separate points of. Also,1 = 1, so 1 A. Therefore Ais dense in C() by the Stone-Weierstrass theorem.

Now we show that is an isometry. Let x A, y = xx. So y = y and y2 = y2 andym = ym. Therefore by the spectral radius formula y = y. Since y = xx, y = |x|2, andso |

x|2 =

y = y = xx

= x2 , proving the isometry. From this,

A is closed in C() and

so A = C().1.9. Theorem (Inverse Gelfand-Naimark). Let A be a commutative C algebra. Let x A besuch that the polynomials in x andx are dense in A. Then we can define an isometric isomorphism : C(Spec (x)) A by

(f) = f x,and we have

f = (f).

Moreover iff() = , then f = x.

Proof. Let be equipped with the Gelfand topology. Then x is a continuous function on withRan(x) = Spec(x). Suppose we have h1 and h2 from such that h1(x) = h2(x). Then alsoh1(x

) = h2(x). By continuity, h1(y) = h2(y) for all y in the algebra generated by polynomials

in x and x, i.e. A. Therefore h1 = h2. Therefore x is one-to-one. Since is compact, x is ahomeomorphism Spec(x). Therefore f f x is an isometric isomorphism ofC(Spec (x))onto C(). By the Gelfand-Naimark theorem, f

x is thus the Gelfand transform of a unique element

in A which we denote f, and f = f. Iff() = , then f x = x and f = x.Remark. This last theorem provides a continuous symbolic calculus for operators as long as theygenerate a commutative C algebra. So, for example, if x is a normal operator then we apply theabove theorem to the algebra generated by x and x, and we get the continuous functional calculusfor normal operators.

1.3 Local Algebras, Idempotents and Projections

Definition. A local Banach algebra is a dense subalgebra A of a Banach algebra A where A is closedunder holomorphic symbolic calculus in A.

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Remark. Note that we need the explicit reference to the completion A, in order to define f(a) fora A because f is required to be holomorphic on the spectrum of the element a, and the spectrumdepends on the whole algebra. However, ifA has a unit, then direct reference to A is not necessary,as shown by the following.

1.10. Theorem. LetA be a local Banach algebra with unit. Let z A be invertible in A. Then z isinvertible in A. Therefore the spectrum of any element is the same in A orA.

Proof. Let z be invertible in A. So the domain of holomorphy of f() = 1 is contained inthe A-spectrum of z, by definition. By definition, A is closed under action of f in its domain ofholomorphy.

Definition. Let A be a local Banach algebra. An idempotent in A is an element x with x2 = x. Ifidempotents x, y satisfy xy = yx = 0, they are said to be orthogonal, written x y. If idempotentsx, y satisfy xy = yx = x, we write x y.

Definition. Let A be a local C algebra. Then two idempotents x, y are said to be orthogonal if, inaddition to the above, we have xy = yx = 0.

Definition. Let A be a local C algebra. An idempotent is called a projection if it is self-adjoint.

1.11. Theorem. The idempotents ofA are dense in the idempotents ofA.

Proof. Let x be an idempotent of A, and let > 0. Choose y in a neighbourhood of x such thaty y2 = x x2 + 2x 2 < . So the spectrum of y is contained completely in an neighbourhood of 0 and 1. Construct the required idempotent by holomorphic calculus. This is not

so clear to me....

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Chapter 2

von Neumann Algebras

2.1 Commutant and Bicommutant

Remark. Now we will introduce von Neumann algebras. These are defined in a concrete sense,

explicitly as subalgebras ofB(H) for a Hilbert space H. Recall the zoo of topologies on B(H).

1. The norm topology or uniform topology.

2. The strong topology is the locally convex topology associated to the family of seminorms

v, v H.

3. The weak topology is the locally convex topology associated with the family of seminorms

| (v,w) |, v, w H.

Definition. A von Neumann algebra is a strongly closed C-subalgebra ofB(H).

Definition. The commutant ofM B(H) is the set Mc = {x B(H) : xy = yx y M}. Clear-ly Mc is a weakly closed subalgebra.

2.1. Theorem (von Neumann). LetM be a C-subalgebra ofB(H), containing the identity. ThenT.F.A.E.

1. M = Mcc.

2. M is weakly closed.

3. M is strongly closed.

Proof. The second clearly follows from the first. To show that the second and third are equivalent,

note the fact that each strongly closed convex set in B(H) is weakly closed.

To show that the last implies the first, let y be a fixed element of Mcc. Let p be the projection ontothe closed subspace of pH = {x : x M} for some fixed H. Clearly py = yp so y pH.Therefore there exists x M such that (y x) < , for each > 0.

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CHAPTER 2. VON NEUMANN ALGEBRAS 12

Take 1, 2, . . . , n H and put = 1n HH. Now yy (M M)cc

.

Apply the construction above to find an x such that (y y x x) < . Then

n

k=1 (y x)k2 < 2.

Therefore y is approximated by x in the strong topology, and so by hypothesis y M. ThereforeMcc M. The opposite inclusion is obvious.

2.2. Theorem (Kaplansky Density). Let A be a C-subalgebra of B(H) with strong closure M.Then the unit ball A1 of A is strongly dense in the unit ball M1 of M. If 1 A, then the unitarygroup ofA is strongly dense in the unitary group ofM.

Proof. See [Kap51].

2.2 Factors

Definition. Let M be a von Neumann algebra. Then the center of M is Z(M) = M Mc.

Definition. IfZ(M) = {1 : C}, then M is called a factor.

Remark. A factor is a kind of algebraic counterpart of an irreducible representation. The factors play

an important role in the classification of von Neumann algebras.

2.3 The Trace

Definition. Let A B(H), A 0. The trace ofA is

Tr (A) =i

(vi, Avi) [0, ].

IfTr (A) < then A is called trace class.

Definition. Consider the family of seminorms i : A |Tr (ABi) |, for {Bi} the set of trace classoperators. The topology associated to this family is called the -weak topology or the ultra-weak

topology.

Remark. Choosing a basis we see that every functional A Tr (AB) can be written as A (vi, Awi), so the ultra-weak topology is stronger than the weak topology. However, these two

topologies coincide on the unit ball ofB(H).

Definition. A bounded functional on a von Neumann algebra M is called normal if for eachbounded monotone increasing net {Ai} in Msa with limit A, the net {(Ai)} converges to (A).The set of normal functionals on a von Neumann algebra M is denoted by M. M is a Banachspace and M

= M. Thus M is called the pre-dual ofM.

2.3. Theorem. Let be a bounded functional on a von Neumann algebra M. Then T.F.A.E.

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CHAPTER 2. VON NEUMANN ALGEBRAS 13

1. is normal.

2. is weakly continuous on the unit ball in M.

3. is ultra-weakly continuous.

4. There is a trace class operatorA such that(B) = Tr (AB) for all B M.

Proof. See [Ped79].

2.4. Theorem. Let be a positive normal functional on a von Neumann algebra M. Then thereexists a positive trace class elementA B(H), such that(B) = Tr (AB) for all B M. Further-more, = Tr (A).

Proof. See [Ped79].

2.4 Hyperfinite Algebras

Definition. A von Neumann algebra M is called hyperfinite if there exists an increasing sequenceof finite-dimensional subalgebras whose union is weakly-dense in M.

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Chapter 3

States and Representations

3.1 GNS Construction

Definition. A linear functional on a C algebra A satisfying (AA) 0 for all A A is called apositive functional.

3.1. Theorem. The following properties hold for positive functionals .

1. IfA is self-adjoint then (A) R.

2. |(AB)|2 (AA)(BB).

3. is continuous w.r.t. the norm topology.

4. (BAB) A (BB).

Proof. These are elementary properties following from the definition.

Definition. The state space for A is defined by

A1+ = { A : 0, = 1} .

3.2. Theorem (Banach-Alaoglu). The unit ball in A, A1 , is compact in the weak-* topology. Andthus so is A1+.

Remark. A1+ is obviously convex. Since it is compact, by the Krein-Milman theorem it is equal tothe convex hull of its extreme points.

Definition. The extreme points ofA1+ are called pure states.

Definition. A representation ofA on B(H) is a C algebra homomorphism : A B(H).

Definition. is called irreducible if the only closed invariant subspaces of H are {0} and H.

Definition. A vector H is cyclic for if the set {(A) : A H} is a total subset ofH.

3.3. Theorem. T.F.A.E.

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CHAPTER 3. STATES AND REPRESENTATIONS 15

1. : A B(H) is irreducible.

2. Every operator that commutes with all of(A) is a multiple of 1. [Schur]

3. Every H is cyclic for.

Definition. 1 and 2 are called unitarily equivalent if there exists an isomorphism V : H1 H2such that 2(A) = V 1(A)V

1 for all A A.

3.4. Theorem. Let 1 and 2 be irreducible and not unitarily equivalent. Then for every boundedT : H1 H2, we have

T 1(A) = 2(A)T A A T = 0.

Proof. Consider TT and use the above result.

Definition. is faithful if one of the following equivalent statements holds.

1. (A) = (B) = A = B.

2. (A) = 0 = A = 0.

3. A (A) is an isomorphism.

3.5. Theorem (Gelfand-Naimark-Segal). Let be a positive functional on A. Then there exists a(cyclic) representation on a Hilbert space, with a cyclic vector such that

(A) = (, (A)) A A.

Furthermore, is unique up to unitary equivalence.

Proof. Let F = {A A : (AA) = 0}. It is easy to show that F is a left ideal. Let (A) =A/F. Note that (A, B) = (AB) defines a positive scalar product on (A). We write (A) forthe projection ofA onto A/F. By completing (A) we get a Hilbert space.

Define the representation by (A)(B) = (AB). It follows from ((B), (A)(B)) =(BAB) that (A) is bounded; (A) A. Thus by continuity it can be extended to all ofH. Set = (1).

All that remains is the proof of uniqueness. Let be another representation of A on H with cyclicvector and such that (A) = ( , (A)) for all A A. The sets (A) and (A) are

each everywhere dense subspaces of H. Thus we can define V by V (A) = (A) for allA A, and V extends by continuity to an isomorphism H H. Now we can use this to prove(A) = V (A)V

1 for all A A.

3.6. Theorem (Gelfand-Naimark). For all A A, A = sup (A).

Definition. Let be the GNS representation for the state . The folium of the representation isthe set of all states of the form

(A) = Tr ((A)) , A A,

for B(H

) trace class and positive.

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3.7. Theorem. The folium of a faithful representation of a C algebra A is weakly dense in A1+.

Proof. See Ref. [Fel60].

Definition. Let be an automorphism of the C algebra A. We say that the positive linear functional

on A is invariant with respect to if((A)) = (A) for all A A.

3.8. Theorem (Unitary Representations). Let be as above. Let(, H, ) be given by the GNSconstruction. Then there exists a unitary operator U such that

((A)) = U (A)U1 A A.

Proof. Define U by using

U (A) = ((A)).

Remark. Note that if we demand U = then U is determined uniquely.

Definition. A subspace of the state space, F A1+, is said to be separating for A if

A A positive and (A) = 0 F = A = 0.

3.9. Theorem. LetA be separable. Then the state

2nn is separating for any dense sequence{n} A1+.

Proof. A is separable, therefore the unit ball ofA is second countable, since it is weak-*-metrizable

and compact. Therefore A1+ is second countable, and the result follows.

Definition. It is customary to say that is faithful if it is separating. This is sensible by the result ofthe next theorem. For each F A1+ we form HF = FH and F = F.

3.10. Theorem. LetF A1+ be a separating family of states forA. Then F is a faithful represen-tation ofA into B(H).

Proof. Let A be positive and A ker(F). Then (A) = ((A), ) = 0 for all F.Therefore A = 0 and ker() = {0}.

3.2 Basic Structure of Representations

Definition. The universal Hilbert space for a C algebra A and the universal representation for Aare defined to be

HA1+, A1+.

Definition. The enveloping von Neumann algebra for A is the strong closure ofA1+(A) in B(HA1+).The enveloping von Neumann algebra is conveniently denoted by Acc. By the following we can alsodenote it as A.

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3.11. Theorem. LetA be a C algebra. Then the enveloping von Neumann algebra ofA is isomor-phic as a Banach space to the second dual ofA.

Proof. Each state of A is a vector state in HA1+ , and therefore a normal state on Acc. Obviously

each element ofA is a linear combination of elements ofA1+. Therefore we can define a map from

A into the pre-dual ofAcc.

Now A is ultra-weakly dense in Acc, so this map is a linear isometry and each in the pre-dual ofAcc will be the image of|A in A. Thus A is the pre-dual ofAcc. Therefore A = Acc.

Definition. Given a (non-degenerate) representation (, H) of a C algebra A, we can find a projec-tion in the enveloping von Neumann algebra, Acc, which takes us down to the image of . In otherwords, this is the projection onto the block(, H) inside the enveloping von Neumann algebra, whichcontains all representation elements. This projection is called the central cover of the representation

(, H). Denote this projection by c().

3.12. Theorem (Central Projections). Let(1, H1), (2, H2) be two representations of a C alge-bra A. These representations are equivalent if and only ifc(1) = c(2). The map (, H) c() isa bijection between nonzero central projections in Acc and equivalence classes of representations ofA.

Proof. For each central projection p = 0 in Acc, we can form a representation for A with the mapA Ap, A A. The Hilbert space for the representation is pHA1+ , and its central cover is p. Thuswe associate a representation with each central projection. Now if (, H) is a representation thenclearly it is equivalent to the representation : A Ac() on c()HA1+.

Remark. It is important to know when a separable C algebra has a representation on a separableHilbert space. In particular, the enveloping von Neumann algebra acts on a generically non-separable

space, and we would like to know how this interacts with representations.

Definition. A von Neumann algebra M is called -finite or countably decomposable if each set ofpairwise orthogonal non-zero projections in M is countable. A projection p on M is called -finiteif pMp is -finite. If M acts on a separable Hilbert space then it is -finite. A partial converse ofthis is true.

3.13. Theorem. A von Neumann algebra M has a faithful normal representation on a separableHilbert space if and only if M is -finite and contains a strongly dense sequence (is countablygenerated).

Proof. Let M B(H), H separable, then M is -finite. Since the unit ball in B(H) is secondcountable for the strong topology, the unit ball in M is second countable and so M is separable inthe strong topology.

Conversely, for each v H define [Mcv] M to be the projection onto the closure of the subspaceMcv. Let {[Mcv]} be a maximal family of these projections, so

[Mcv] = 1. IfM is -finite then

{vn} is countable. Let (A) =

2n (Avn, vn), then is a normal state on M. It is also clear that is faithful. M is countably generated therefore there exists a C algebra A which is separable andstrongly dense in M. H contains a dense separable subspace, so it is separable.

3.14. Corollary. A representation (, H) of a separable C algebra A is equivalent to a separablerepresentation if and only ifc() is -finite in Acc.

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3.15. Theorem. Let (1, H1) and (2, H2) be non-degenerate representations of a C algebra A.Then T.F.A.E.

1. c(1) c(2).

2. (1 2)Acc = cc1 (A) cc2 (A)

3. (1 2)Ac = c1(A)

c2(A)

4. There are no equivalent subrepresentations of(1, H1) and(2, H2).

Proof.

1=2

ker(1 2) = A(1 c(1) c(2))

= ((1 2)

A)cc

= Acc

(c(1) + c(2))= ((1 2)A)

cc = 1(A)cc 2(A)

cc.

2=3 Follows from von Neumanns bicommutant theorem.

3=4 Assume there exists an isometry u : H1 H1. By definition of equivalence uu 1(A)c,

uu 2(A)c, and u(2(A)uu

)u = 1(A)uu for all A A. Now

(1 2)(A)u = (1(A) + 2(A))u

= u1(A) = 2(A)u

regarding u as an element ofB(H1 H2)

= u(1(A) + 2(A))

= u(1 2)(A)

=u ((1 2)(A))c

By assumption ((1 2)(A))c B(H1) B(H2). So u = 0.

4=1 If 1 does not hold then consider subrepresentations with central cover c(1)c(2).

Definition. Representations satisfying the properties of the previous theorem are called disjoint rep-

resentations.

Definition. A non-degenerate representation (, H) of a C algebra A is called a factor represen-tation when (A)cc is a factor. Note that (, H) is a factor representation if and only if c() is aminimal projection in the center ofA. Two factor representations are either equivalent or disjoint.

Definition. Let (, H) be a representation of a C algebra A. If K H is a linear subspace with(A)K K, then K is called reducing for . Representations satisfying the conditions of thefollowing theorem are called irreducible.

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3.16. Theorem (Irreducible Representations). Let(, H) be a nonzero representation of a C al-gebra A. Then T.F.A.E.

1. There are no non-trivial reducing subspaces for.

2. (A)c = {1}.

3. (A) is strongly dense in B(H).

4. Each non-zero v H is cyclic for(A).

5. (, H) is equivalent to a cyclic representation associated with a pure state.

Proof. The proof is straightforward computation. See [Ped79].

3.17. Corollary. Two irreducible representations (1, H1) and(2, H2) of A are either disjoint orequivalent.

Proof. If they are not disjoint then they have equivalent subrepresentations by a previous result. But

irreducible representations have only trivial subrepresentations by the above.

3.18. Theorem (Repelling Representations). Let and be pure states of a C algebra A. If < 2 then (, H) and(, H) are equivalent. If they are equivalent, then = (u u)for some unitary u A.

Proof. Assume (, H) and (, H) are not equivalent. Then they are disjoint, c() c().Now (c()) = 1 and (c()) = 1, so (c()) = 0 and (c()) = 0, so ( )(c() c()) = 2.

To prove the second part, assume the representations are equivalent. Then for every A A we have

(A) = ((A), ) for some unitary H.

Let u be the unitary which takes to , (u) = . Then

(A) = ((A)(u), (u))

= ((uAu), )

= (uAU).

Remark. Previously we introduced the idea of the Gelfand transform of a commutative Banach al-

gebra. This was a map from algebra elements to functions h, h : C. The generalization ofthis to the non-commutative case is connected to representation theory.

Definition. Let Irr(A) be the set of irreducible representations ofA. Define the spectrum ofA, A,to be the set of equivalence classes of irreducible representations of A.

Remark. When A is commutative, all the irreducible representations are one-dimensional. Then Ais nothing but the set of non-zero complex homomorphisms of A, which is the Gelfand transform ofA. One approach to the non-commutative case is through the so-called decomposition theory. Thebasic object in decomposition theory is the atomic representation.

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Definition. Let A be a C algebra. For each t A choose pure state t with representation (t, Ht).Define the atomic representation to be (a, Ht) with

a = t At Ha = t AHt.

Remark. The above definition involves a choice, but the equivalence for different choices is easy to

show, so the atomic representation is essentially unique.

3.19. Theorem.

a(A)cc =t A

B(Ht).

Proof. By a previous theorem t(A) is strongly dense in B(Ht) for each t. Therefore t(A)cc =

B(Ht). The ts are mutually disjoint so the result follows.

Remark. The decomposition theory proceeds by making A into a measure space, beginning with theso-called D-Borel structure. The first major result is a classification of equivalent representationsaccording to the measures which are associated to them via their states. See [Kad57]. The converse

construction, building representations up from their measures leads to the theory of the direct integral.

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Chapter 4

Structure Theory II

4.1 Weights and Traces

Definition. Let A be a C algebra. A weight on A is a function : A+ [0, ] such that

1. (A) = (A) A A+, R+.

2. (A + B) = (A) + (B) A, B A+,

where A+ is the set of positive elements ofA.

Definition. A weight is said to be densely defined if the set A+ = {A A+ : (A) < } is densein A+.

Definition. Let M be a von Neumann algebra. We say that is semi-finite ifM+ is weakly densein M. For von Neumann algebras this coincides with the notion of -finite.

Definition. A weight on a von Neumann algebra M is called -normal if there exists a sequence{n} of sequentially normal positive functionals on M such that (x) =

n(x) for all x M+.

Definition. is called lower semi-continuous if for each R+ the set {A A+ : (A) } isclosed.

Definition. A trace on a C algebra A is a weight such that (uAu) = (A) for all A M+ and

u unitary.

4.1. Theorem (Radon-Nikodym). Let and be normal functionals on a von Neumann algebraM such that 0 . Then for each C with Re 1/2 there is an elementh M1+ suchthat

= (h) + (h).

If is faithful then h is unique.

Proof. Let N =

(h) + (h) : h M1+

. N is compact and convex since M1+ is convex

and ultra-weakly compact. N is a subset of the pre-dual of M. If N, then there is an element

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CHAPTER 4. STRUCTURE THEORY II 22

in the self-adjoint part of M, Msa, say a Msa, and a t R such that (a) > t, N(a) t. Leta = a+ a and take h = [a+]. Then

(a+) (a+ a) > t 2Re (a+) (a+) .

If is faithful and if = (k) + (k) for some k Msa, then since

( + )(h k)2 = h(h k) + (h k)h k(h k) (h k)k,

we have

2Re ((h k)2) = (h k) (h k) = 0,

= h = k.

Definition. Let M be a von Neumann algebra on a separable Hilbert space. We have the followingnomenclature.

M is called finite if it admits a faithful, normal, finite trace.

M is called semi-finite if it admits a faithful, normal, semi-finite trace.

M is called properly infinite if it does not admit a non-zero, normal, finite trace.

M is called purely infinite if it does not admit a non-zero, normal, semi-finite trace.

4.2. Theorem (First Decomposition). Let M be a von Neumann algebra. Then M has a uniquedecomposition

M = M1 M2 M3,

where

M1 is finite,

M2 is semi-finite but not properly infinite,

M3 is purely infinite.

Proof. Let be a normal trace on M, so is weakly lower semi-continuous. Therefore N ={x M : (xx) = 0} is a weakly closed ideal ofM. ThereforeN = (1 p)M for some centralprojection p M, and is faithful on pM.

Also, the weak closure of M = {x M : (x) < } is an ideal of M, so there is a centralprojection q such that is semi-finite on qM and purely infinite on (1 q)M. Therefore is faithfuland semi-finite on pqM.

Let {n, pn} be a maximal family of normal finite traces n and pairwise orthogonal projections pnsuch that n is faithful on pnM. M is -finite [Ped79] so {n, pn} is countable. Define

(x) = 2nn(1)1n(pnx).

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Ifp =

pn then is faithful, normal, and finite on pM, and by maximality of{n, pn}, (1 p)Mis properly infinite.

Let {n, qn} be a maximal family of normal, semi-finite traces n and pairwise orthogonal projec-tions qn 1 p such that n is faithful on qnM. Let (x) =n(qnx), q = qn. Then q p, is faithful, normal, and semi-finite on qM. (1 q p)M is purely infinite by maximality of{n, qn}.

4.3. Theorem. If M is a semi-finite von Neumann algebra on a separable Hilbert space, then Mc

is semi-finite.

Proof. The proof requires a somewhat technical result. See [Ped79].

4.2 Types

Definition. Let p and q be projections in a C algebra A. If there exists a partial isometry v Asuch that vv = p and vv = q, we say that p is equivalent to q, writing p q. Recall that u isa partial isometry if uu (and thus uu) is a projection. This coincides with the previous notion ofprojection equivalence.

Remark. As an example, suppose A = B(H), then two projections are equivalent if and only if pHand qH have the same dimension. Thus the equivalence classes of projections on a von Neumannalgebra are a sort of generalized dimension set.

Definition. Let x be in Msa. The central cover ofx, c(x), is the infimum of all z Zsa with z x.It exists because Zsa is a complete lattice.

Definition. A projection p is called abelian ifpAp is a commutative algebra.

Definition. A von Neumann algebra A is called type I if there is an abelian projection p M withc(p) = 1.

4.4. Theorem. LetM be a von Neumann algebra of type I, on a separable Hilbert space, and let pbe an abelian projection with c(p) = 1. Then there is a faithful, normal, semi-finite trace on Mwith (p) = 1.

Proof. pMp is a commutative von Neumann algebra on a separable Hilbert space, thereforepMp =L (T) for some locally compact, Hausdorff, second countable measure space T with measure .

Take any finite measure on T equivalent to as on pMp. Normalize to (p) = 1. extends to anormal semi-finite trace on M, and since c(p) = 1, is faithful on M.

Definition. A von Neumann algebra M is said to be homogeneous of degree n if1 =n

i=1pi, forsome {pi} a set ofn orthogonal, equivalent, abelian projections.

4.5. Theorem. LetM be a von Neumann algebra of type I on a separable Hilbert space. Then Mhas a unique decomposition

M =

Mn, 1 n ,

with Mn

homogeneous of degree n.

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Proof. First we show that every type I M contains a nonzero, homogeneous, central summand. Let{qi} be a maximal family of orthogonal abelian projections in M with c(qi) = 1. The family isnon-empty by definition of type I. Let z = 1 c(1

qi). If z = 0, then there would be an

abelian projection q 1

qi with c(q) = c(1

qi) = 1, contradicting the maximality of

{qi}. Therefore

c(z zqi) = zc(1 qi) = 0, but z = 0. Therefore z = zqi, and so Mz ishomogeneous.Let {z(n)j } be the maximal family of orthogonal central projections each of which is a sum of northogonal, equivalent, abelian projections {pji}. c(pji) = zj and pi =

j pji is abelian for each

1 i n. Let en =

j zj. Then c(pi) = en, so {pi} is a family of orthogonal, equivalent, abelianprojections since equivalence of projections follows from equality of their central covers.

Now

ipi =

ij pji =

j zj = en, so Men is homogeneous of degree n. For n = m, enem = 0by the well-defined-ness of the degree of homogeneity.

Since {zj} is maximal, M(1

en) contains no homogeneous central summand. But M(1

en)

is clearly of type I, which is a contradiction.

4.6. Corollary. Let M be a factor of type I. Then M is isomorphic to B(H) where dim H is thedegree of homogeneity ofM.

Proof. Let p be a non-zero abelian projection in M. Since M is a factor, p is minimal and c(p) = 1.Let be a normal state with (p) = 1. Then is pure and so (, H) is irreducible. Thus (M) =B(H). The degree of homogeneity of (M) is obviously equal to that for M.

4.7. Lemma. Let M be a von Neumann algebra of type I on a Hilbert space H. Then Mc isisomorphic to a von Neumann algebra with abelian commutant.

Proof. Let p be an abelian projection with c(p) = 1. Then Mc

= Mc

p, and (Mc

p)

c

on pH ispMp.

4.8. Lemma. LetM be a commutative von Neumann algebra on a Hilbert space H. Then Mc is oftype I.

Proof. Let q be a non-zero projection in Mc and choose a unit vector v qH. Let p be the cyclicprojection on the closed subspace [Mv]. Then p Mc and p q.

Now Mp is commutative and has a cyclic vector, so it is maximal commutative on pH. ThereforeMp = (Mp)c = pMcp. Therefore p is an abelian projection and Mc is type I.

4.9. Theorem. LetM be a von Neumann algebra. Then T.F.A.E.

1. M is of type I.

2. Mc is of type I.

3. M is isomorphic to a von Neumann algebra with abelian commutant.

4. Mc is isomorphic to a von Neumann algebra with abelian commutant.

Proof. 1 =4, 2=3 follow from the first lemma.4 =2, 3=1 follow from the second lemma.

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Definition. M is said to be type II if it is semi-finite, but contains no non-zero abelian projections.

Definition. M is said to be type III if it is purely infinite.

Remark. Notice that M is finite if it is homogeneous of degree n with n < . It is properly infinite

if n = . We thus subdivide type I into type In, 1 n . If M is type II, then we say it istype II1 if it is finite and type II if it is properly infinite. The following theorem is a restatementof this classification.

4.10. Theorem (Second Decomposition). LetM be a von Neumann algebra on a separable Hilbertspace. Then M has a unique decomposition into central summands of each type,

M = MI

MI

MII1

MII

MII I.

Remark. We would like to extend the notion of type to C algebras which are not necessarily vonNeumann algebras. This is what we do in the following.

Definition. Let A be a C algebra, and let A A. The hereditary algebra generated by A is thenorm closure ofAAA.

Definition. A positive element A A is called abelian if the hereditary algebra generated by A is acommutative algebra.

Definition. A is a type I C algebra if each non-zero quotient of A contains a non-zero abelianelement. IfA is actually generated by its abelian elements, then we say A is of type I0.

Definition. IfA contains no non-zero abelian elements, then we say it is antiliminary.

Remark. A von Neumann algebra of type I is not in general a C algebra of type I. As an example,

let M = B(H) for an infinite-dimensional H. Let K(H) denote the compact operators. ThenB(H)/K(H) contains no abelian elements.

4.11. Lemma. Let A be a positive element of a C algebra A. Then A is abelian if and only ifdim (A) 1 for every irreducible representation (, H) ofA.

Proof. Suppose A is abelian and (, H) is an irreducible representation. Then (A) is abelian in(A), so (A)B(H)(A) is commutative and so dim (A) 1.

Conversely, let A be positive in A and suppose dim (A) 1 for each irreducible representation .Then AAA is commutative in the atomic representation, which is faithful. So A is abelian.

4.12. Lemma. LetA be a C algebra acting irreducibly on a Hilbert space H such thatAK(H) =0. Then K(H) A and each faithful irreducible representation of A is unitarily equivalent to theidentity map.

Proof. A K(H) = 0, therefore there is a finite-dimensional projection in A K(H) and a one-dimensional projection p A. If is a unit vector in pH then for any H there is an A A suchthat A = ; A acts irreducibly on H. Therefore ApA is the projection on C. Thus A contains allthe one-dimensional projections, so K(H) A.

Furthermore, let be a pure state on A. Then (, H) is faithful. |K(H) is non-zero since (, H)is faithful, so it is a state for K(H). Since the dual of K(H) is the set of trace class operators,(x) = (x,), some H, for all x K(H).

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But the extension of a state from an ideal to the whole algebra is unique, so on A is a vector state(A) = (A,). Thus any such determines a cyclic representation with a cyclic vector and anysuch is equal to the identity representation on cyclic vectors, and thus unitarily equivalent to theidentity representation.

4.13. Theorem. LetA be a C algebra of type I. Then K(H) (A) for each irreducible repre-sentation (, H) ofA.

Proof. (, H) is irreducible. By the first lemma, for A abelian in A with norm 1, there is a one-dimensional projection on H. By the second lemma K(H) (A).

4.14. Corollary. LetA be a type-I0 C algebra. Then for every irreducible representation (, H) ofA, (A) = K(H).

Proof. (A) is generated by its abelian elements, (A) K(H). But K(H) (A).

Definition. A C

algebra A is called liminary if(A) = K(H) for each irreducible representation(, H) ofA. Thus each type-I0 C algebra is liminary, but the converse is false.

Remark. The following are useful properties which we state without proof.

4.15. Theorem. A liminary C algebra is of type I.

Proof. See [Ped79].

Definition. IfA is a C algebra, we define a composition series to be a strictly increasing family ofclosed ideals {I} indexed by [0, ], a segment of the ordinals, with I0 = 0,I = A, and suchthat for each limit ordinal we have

I = norm closure

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Chapter 5

Matrices

5.1 Inductive Limits

Definition. Define a directed set I to be a partially ordered set such that if, I then there exists I with < , < .

Definition. Let I be a directed set. Let X be locally convex spaces with varying in I. LetX =

IX

; X is a locally convex space. Suppose that < if and only if X X and thatthe inclusion is continuous. Suppose also that for any convex V X, V is a nbhd. ofU X if andonly if I V X is a nbhd. ofU X. When all the above conditions hold we say that X isthe inductive limit ofX.

Definition. When the X are Banach spaces, in particular Banach algebras, the inclusions in the def-inition of inductive limit are bounded linear maps. When these inclusions also satisfy lim supalpha < for all , the system is called a normed inductive system, and the limit X is called a normed induc-tive limit. This extra uniformity condition implies that x = lim sup

(x), for x X X,is a seminorm on X. Quotienting by elements of zero seminorm and completing gives a Banachspace, which will also be called the inductive limit ofX, and again we will write X = lim X

.

5.2 Glimm Algebras

Remark. Now we construct some antiliminary algebras which are interesting both as examples of

non-intuitive algebras and as physical fermion algebras. These are the Glimm algebras. We need the

notion of inductive limits for locally convex spaces.

Definition. Let Mm be the C algebra of m m matrices, identified with B(Hm). Suppose i :

Mm Mn is a morphism of Mm into Mn with i(1) = 1. Let d = Tr

i(v(m)11 )

, where v(m)11 is the

matrix with 1 in the (1, 1) place and zeroes elsewhere. We have md = n.

Let {s(n) : n N} be a sequence of natural numbers, greater than one. Let s(n)! =n

k=1 s(k).Then consider the inductive system

Ms(1) Ms(2)! Ms(n)! ,

27

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CHAPTER 5. MATRICES 28

with the inclusion map i. The inductive limit M = Ms(n)! is not necessarily norm complete, butits completion, A, is a C algebra which is called the Glimm algebra of rank {s(n)}.

Definition. The fermion algebra is the Glimm algebra for which s(n) = 2 n N.

5.1. Theorem. Every Glimm algebra is a separable, simple (contains no non-zero closed ideals)C algebra and has a unique tracial state.

Proof. Each Ms(n)! is separable and M is dense in A, so A is separable. If is a non-zeromorphism ofA, then |Ms(n)! is an isometry for each n. Therefore M is an isometry and is anisometry. Thus A is simple (every morphism is an isometry).

Let n be the normalized trace on Ms(n)!. Then n+1 i = n, so there is a unique tracial state onM, and so it is tracial on A. Conversely if is tracial on A then from the uniqueness of thetrace on B(Hs(n)!) = .

5.2. Theorem. There exists a factor of type II1.

Proof. M = (A)cc has a non-zero finite normal trace, the extension of . ker() is a central

projection, so we can assume that is faithful on zM for some central projection z; is faithful onM so z = 0.

Since is the unique tracial state on A, the center ofzM is trivial, so zM is a factor. This factoris finite but not finite-dimensional, so it is of type II1.

Remark. Let F denote the fermion algebra. For each [0, 1/2] we can construct a state on F asfollows.

Let {n} be a sequence of convex combinations each of length 2, i.e. 1 = 2, 21 + 22 = 2, etc.

Note that Ms(n)! = Ms(1) Ms(n), so each element ofMs(n)! can be written

x = x(1) x(n), x(k) Ms(k).

Let

(x) =n

k=1

s(k)i=1

ki x(k)ii

.This extends to a unique state on F. It is called the product state on F. Note that the tracial state onF is the product state with ni = 1/2 for all i 2, n N. note that ifx Ms(n)! and y Ms(n)!

c,

then (xy) = (x)(y).Remark. For each [0, 1/2] we choose = {n} to be the sequence of convex combinationsn1 = ,

n2 = 1 , for all n N.

Let be the product state associated with and let (, H, ) be the cyclic representation of Fassociated with . We already know that

0(F)cc = B(H0),

1/2(F)cc = factor of type II1.

5.3. Theorem. Each product state of a Glimm algebra is factorial, i.e. gives rise to a factor repre-

sentation.

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Proof. Let be a product state on A of rank {s(n)}. Let z Z((A)cc). By Kaplanskys

density theorem there exists a sequence {yk} in A with yk z and (yk) z weakly. LetUs(n)! denote the unitary group ofMs(n)!. This is a compact group with Haar measure du. Let

zk = Us(n)!

duuyku

, for fixed n.

zk commutes with Ms(n)!, and for each u Us(n)! we have (uyku) (u)z(u) = z weakly.Therefore (zk) z weakly by the Lebesgue dominated convergence theorem.

So for every x, y Ms(n)! we have

(zx, y) = lim((zk)x, y)

= lim (yzkx)

= lim (zk)(yx)

= (z, ) (x, y) .

This holds for any n so z = (z, ) 1, so that z is a multiple of 1. Therefore (A)cc

is a

factor.

Remark. Let be the group of permutations ofN which leave all but a finite number of elementsfixed. For a t we define a unitary operator by

ut : H2n H2n

ut(v1 vn) = vt(1) vt(n).

Fix a sequence{

un} {

ut

: t

}

such that the permutation tn

corresponding to un

satisfies

tn(i) > n i n. It is clear that (utxut ) = x F, t .

5.4. Lemma. Letx F, then (unxun) (x)1 weakly.

Proof.

(zunxu

ny) = (unxu

nzy)

= (unxun)(z

y)

for all x,y,z M2k

and n k. By continuity (z

unxu

ny) (x)(z

y) for all x,y,z F.Therefore ((unxun)y, z) (x) (y, z). Now {(unxun)} is bounded and weakly conver-

gent on a dense set of vectors. Therefore (unxu) (x)1 weakly.

5.5. Lemma. Let be a positive functional on F with a normal extension to (F)cc

for some

[0, 1/2]. Suppose also that (utxut ) = (x) x F, t . Then is a scalar multiple of .

Proof. is weakly continuous on bounded sets in (F) so by the lemma (x) = (unxun) (x)(1), therefore = (1).

5.6. Theorem. The von Neumann algebras M = (F)cc

, for 0 < < 1/2 , are factors oftype III.

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Proof. Each M is a factor by a previous result. Let be a normal, faithful, semi-finite trace on M.Thus there exists a unique positive operator h on H such that (x) = (hx). Now h is unique and

((ut )h(ut)x) = (h(ut)x(u

t ))

= ((ut)x(u

t ))= (x),

so (ut )h(ut) = h for all t .

Pick > 0 and put () = (h( + h)1). h( + h)1 commutes with all (ut) so we have((utxu

t )) = ((x)). Therefore = (1).

Choose x (M)+ such that (x) < and (x) = . This is possible since is semi-finite.Then

(1)(x) = (x) = (h( + h)1x) (x).

Therefore (1) < as 0, and for any x (M)+ we have

(x) = lim (1)(x)

= lim (h( + h)1x)

= (x).

Therefore (x) = . But we know that is not a trace when = 1/2, so there is a contradiction.Therefore M is of type III.

5.3 Matn (A)

Definition. Let A be a Banach algebra. Let Matn (A) denote the n n matrix algebra over A. ThenMatn (A) can be made into a Banach algebra in a number of equivalent ways.

Definition. Define GLn(A) to be the group of invertible elements in Matn (A) which are congruentto 1n mod Matn (A).

Remark. IfA has a unit, then GLn(A) is isomorphic to the group of invertible elements ofMatn (A).

Definition. Let A be a C algebra. Define Un(A) to be the group of unitary elements in Matn (A+)which are congruent to 1n mod Matn (A).

Definition. Mat (A) is the inductive limit Mat (A) = lim Matn (A) with the obvious choiceof isometric inclusions.

Definition. GL(A) = lim GLn(A).

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Chapter 6

Automorphism Groups

6.1 Automorphisms and Invariant States

Remark. It is necessary to fix some notation and basic ideas from the theory of locally compact

groups. Let M(G) denote the Banach space of bounded complex Radon measures on a locallycompact group G, identified with C0(G)

. M(G) possesses convolution and involution but is not in

general a C algebra;f(s)d( )(s) =

f(ts)d(s)d(t), f C0(G)

f(s)d(s) =

f(s1)d(s)

, f C0(G).

Definition. A unitary representation (u, H) ofG is a homomorphism t ut ofG into the unitarygroup ofB(H), which is continuous in the weak topology on B(H). Note that the weak, ultra-weak,and strong topologies coincide on the unitary group of B(H). The representation is called uniformlycontinuous if it is continuous in the norm topology for B(H).

Definition. The universal representation (u, Hu) ofL1 (G)dg is the direct sum of all non-degeneraterepresentations ofL1 (G)dg. The group C algebra ofG, C(G), is the norm closure ofu(L

1 (G) dg)in B(Hu).

Remark. By a representation of a Banach algebra, here we mean an involution-preserving homomor-

phism into B(H), for a Hilbert space H.

Definition. For each M(G) and f L2

(G) dg the convolution f is in L2

(G)dg. Definethe map

: M(G) B(L2 (G) dg)

()f = f.

It is easy to check that it is a representation ofM(G). We call it the regular representation.

Remark. We can identify the points of G with the point measures s, s G. The restriction of tothe point measures is thus identified with the unitary representation s s ofG on L2 (G)dg givenby

(sf)(t) = f(s1t), f L2 (G) dg

31

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CHAPTER 6. AUTOMORPHISM GROUPS 32

Definition. The group von Neumann algebra for G, M(G), is the weak closure of(L1 (G) dg) inB(L2 (G)dg).

Definition. Suppose A is a C algebra with A L (G) dg and that A is invariant under lefttranslation. We say that a state m on A is a left-invariant mean ifm(sf) = m(f) for all f A.

Definition. If there exists a left-invariant mean on L (G)dg then we say that G is amenable.

Definition. Let U C(G) denote the algebra of bounded uniformly continuous functions on G. LetC(G) denote the algebra of bounded continuous functions on G.

6.1. Theorem. LetG be a locally compact group. Then T.F.A.E.

1. G is amenable.

2. There exists a left-invariant mean on C(G).

3. There exists a left-invariant mean on U C(G).

4. There exists a state on L (G)dg, m , such thatm( f) = (G)m(f) for each M(G)andf L (G) dg.

Proof. See [Ped79].

6.2. Theorem. G is amenable if and only if the regular representation is faithful on C(G).

Proof. See [Ped79].

Remark. When we speak of the Haar measure on G, we mean, for example, the left Haar measureso that d(ts) = ds. There is also a right Haar measure, and it is connected to the left Haar measureby the modular function : G R+, d(st) = (t)ds, d(s1) = (s)1ds.

Definition. A C-dynamical system is a triple (A, G , ) with a C algebra A, a locally compactgroup G, and a continuous homomorphism : G Aut(A). Aut(A) is equipped with thetopology of pointwise convergence, so for each A A (A) : G Aut(A) given by t t(A)is continuous. When G and A are separable we call this a separable dynamical system.

Remark. When M is a von Neumann algebra we consider the topology of pointwise weak conver-gence on Aut(M). This is equivalent to pointwise ultra-weak convergence and pointwise strongconvergence since these coincide on the unitary group of M and the unitary group is stable underAut(A) and generates M linearly.

Definition. A W-dynamical system is a triple (M, G , ) with : G Aut(M) continuous inthe topology of pointwise weak convergence.

Definition. A covariant representation of a C-dynamical system (A, G , ) is a triple (,u, H)where (, H) is a representation of A, and (u, H) is a unitary representation of G, and we have(t(A)) = ut(A)u

t .

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Remark. Let (A, G , ) be a C-dynamical system. Let K(G, A) be the space of continuous func-tions of compact support from G to A. Define involution and convolution on K(G, A) by

y(t) = (t)1t(y(t1)),

(y z)(t) = y(s)s(z(s1t))ds.Define y1 =

y dt. Then K(G, A) is a normed algebra with isometric involution. Denote its

completion by L1 (G)A.

Let A A and f L1 (G) dg. We can define A f L1 (G) A such that (A f)(t) = Af(t). Thespan of such elements is dense in L1 (G)A.

6.3. Theorem. If(,u, H) is a covariant representation of(A, G , ) , then there exists a non-degeneraterepresentation ( u, H) ofL1 (G)A such that

( u)(y) = (y(t))utdt for all y K(G, A).Moreover, the correspondence (,u, H) ( u, H) is a bijection onto the set of non-degeneraterepresentations ofL1 (G)A.

Proof. See [Ped79].

Definition. The universal representation (u, Hu) ofL1 (G)A is the direct sum of all non-degeneraterepresentations ofL1 (G)A.

Definition. The crossed product of (A, G , ) is the norm closure of u(L1 (G) A) in B(Hu). It isdenote by G A.

Remark. Now we will introduce some ideas due to Stormer which have direct physical relevance.

See Refs. [Sto69, Sto67, DKS69].

Definition. Let Conv (W) denote the smallest convex subset of the vector space V W containingW. We say that G is represented as a large group of automorphisms of A if, for each G-invariantstate , we have

weak closure {(Conv (G(A)))}

(A)

c = , A A.

Definition. We say that the C-dynamical system (A, G , ) is asymptotically abelian if there is anet G such that

At(B) t(B)A 0 as t in .

We say that it is weakly asymptotically abelian if we have

(At(B) t(B)A) 0 as t in , for any A.

6.4. Lemma. Let(A, G , ) be a weakly asymptotically abelian C-dynamical system. Then G is alarge group of automorphisms.

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Proof. Let be an invariant state and A A. Let z denote any weak limit of the bounded net{(t(A)) : t }. Then clearly z is in the weak closure of {(Conv (G(A))). Moreover, forany B A and in the pre-dual of(A)

cc, we have

(z(B) (B)z) = lim ((t(A)B Bt(A)))

= 0

Therefore z (A)c. So G is large.

Definition. If is a G-invariant state ofA, then we say that is asymptotically multiplicative withrespect to the net if

(t(A)B) (A)(B) as t in .

Such states are also called strongly clustering or strongly mixing.

6.5. Theorem. Let(A, G , ) be a weakly asymptotically abelian C-dynamical system and considera G-invariant state on A, with covariant cyclic representation (, u, H, ). Then T.F.A.E.

1. is asymptotically multiplicative.

2. is an extreme point of the set of G-invariant states on A , and for each A A the net{(t(A)) : t } is weakly convergent to (A) 1 in B(H).

3. The net

ut : t

is weakly convergent in B(H) to the one-dimensional projection on

C.

Proof. See [Sto69, Sto67, DKS69].

6.6. Corollary. Let (A, G , ) and be as above. If is a factor state, then it is asymptoticallymultiplicative.

6.7. Theorem. Let (A, G , ) be a C-dynamical system with G a large group of automorphisms.Let be a G-invariant factor state with cyclic covariant representation (, u

, H, ). LetM =(A)

cc, and let be the vector state on B(H) determined by . Then

1. M is finite is a trace on M.

2. M is semi-finite but infinite is a trace on Mc, but not on M.

3. M is type III is not a trace on Mc.

6.8. Theorem. Let (A, G , ) be a weakly asymptotically abelian C-dynamical system with G a-belian, and let be a G-invariant factor state ofA with cyclic covariant representation (, u, H, ).LetM = (A)

cc, and let be the vector state on B(H) determined by . Then

1. M = C 1 is multiplicative.

2. M = B(H

), dim H

= is a pure state but not multiplicative.

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3. M is type II1 is a trace but not multiplicative.

4. M is type II is a trace on Mc but is neither pure nor a trace.

5. M is type III is not a trace on Mc.

Remark. In the case that G is abelian we have a zoo of results. These results can be thought ofas harmonic analysis on operator algebras. They will lead us to the Tomita-Takesaki theory, the

introduction of complex function theoretic techniques for one-parameter groups (KMS states, etc.),

and a classification of factors of type III.

Remark. First we want to associate subsets of the dual group of G to subspaces of a Banach spaceX, when G acts as isometries on X. When X is a C algebra and G acts as automorphisms we willbe able to construct a spectral measure on the dual group with () corresponding to the supportprojection of the subspace associated to . When this happens we will be able to construct a unitaryrepresentation ofG which, under certain conditions, is covariant for the automorphism representationofG.

Definition. Let X and X be two Banach spaces in duality via a bilinear form , . This means

Ifx X then x, X.

If X then , X.

The maps x x, and , are isometries ofX and X onto weak-* dense subspacesofX

and X respectively.

Let B(X) and B(X) denote the bounded linear operators which are continuous in the (X, X)and (X, X) topologies. Note that ifU B(X) then U B(X) if and only ifUT B(X).

A representation of a locally compact G on X is a (X, X) continuous homomorphism t t ofG onto the group of invertible elements in B(X). We say that is an integrable representation iffor each M(G) there is a (necessarily unique) B(X) such that

(x), =

t(x), d(t), x X, X.

Note that T is integrable whenever is integrable.

6.9. Lemma. Let X be a Banach space and X = X. Leth : t t be a homomorphism of a

locally compact group G into the group of invertible isometries on X such that t t(x), x X,

is norm continuous. Then h is an integrable representation ofG on X.Proof. See the appendix of [Ped79].

Definition. Let G be a locally compact abelian group and let denote its dual group. Denote theunit in by . For t G and let (t, ) denote the value of at t and write (t) = (t, )d(t)for each M(G), i.e. the inverse Fourier transform.

Let K1(G) be the dense ideal of L1 (G)dg consisting of functions such that f has compact supportin . Let X and X be as introduced previously and let be an integrable representation ofG on X.For each open we define the spectral R-subspace

R() = (X, X) closure in X of the linear subspace f(x) : x X, f K1(G), suppf .

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For each closed the spectral M-subspace is

M

() =annihilator of

R

(|).In other words,

x M()

x, Tf()

= 0 X, f K1(G) with suppf |.

6.10. Theorem. LetR andM be as above. Then

1. If1 2 then R(1) R(2).

2. If1 2 then M(1) M(2).

3. The -closure ofi R(i) is equal to R(ii).4. iM(i) = M(ii).

5. If then R() M().

6. If then M() R().

7. If = ii = ii then M() = iR(i).

8. If = ii = iinti then R() = -closure

i M

(i).

9. R() = M() = {0}; R() = M() = X.

Proof. See [Ped79].

Definition. From the fourth point above, there exists a smallest closed set such that M() =X. We call the Arveson spectrum of and denote it Spec().

6.11. Theorem. Let be an integrable representation of G on X. For each , T.F.A.E.

1. Spec().

2. R() = 0 for every nbhd. of.

3. There exists a net{xi} in the unit sphere ofX such thatt(xi) (t, )xi 0 uniformly oncompact subsets ofG.

4. For every M(G) we have || .5. For every f L1 (G) dg we have |f()| f.6. Iff L1 (G) dg andf = 0 then f() = 0.

Proof.

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1=2 IfR() = 0, then Spec() \. Conversely if Spec() then there exists an opennbhd. of with Spec() = . Therefore R() = 0.

2=3 This follows from the following technical lemma which we do not prove: For , > 0,and Ka compact subset ofG, there exists a compact nbhd. of such that t(x) (t, )x 0 there is a compact K G such that |(G\K)| < . Assume xi X andt(xi) (t, )xi < for all t K; then

|()| = ()xi=

(t, )xid(t)

(t(xi) (t, )xi)d(t)

+

t(xi)d(t)

|(K)| + 2|(G\K) + (xi)

+ 2 + .

Therefore () .4=5 Obvious.

5=6 Obvious.

6=2 Let be a nbhd. of. There exists f K1(G) with supp(f) , f() = 1. Then byassumption f(x) = 0 for some x X, and so R() = 0.

6.12. Theorem. Let be an integrable representation of G on X. If A is the commutative Banachalgebra in B(X) generated by f, f L1 (G) dg, then the Arveson spectrum of is homeomorphicto the Gelfand spectrum ofA.

Proof. The dual of the homomorphism : L1 (G) dg A defines a continuous injection :A since is the spectrum ofL1 (G)dg. A is locally compact so is a homeomorphism ontoits image. From the previous proposition then ( A) if and only if Spec().6.13. Theorem (Compact Arveson Spectrum). Let be an integrable representation of G on X.Then T.F.A.E.

1. Spec() is compact.

2. is uniformly continuous, i.e. 1 t 0 as t 0.

Proof.

1=2 Let f K1(G) with f = 1 on an open set containing Spec(). Then f(x) = x, x R(), and since R() = X, f = 1. But then

1 t(x) f t f1 x , x

= 1 t 0 as t 0.

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2=1 Let (f) be an approximate identity for L1 (G)dg.

x t(x)

t(x) x f(t)dt

t 1 f(t)dt x .Therefore f 1 so the Banach algebra generated by (L

1 (G) dg) contains the identity, andso Spec() is compact by the previous theorem.

6.14. Theorem (Stone). Lett ut be a unitary representation of an abelian group G on a Hilbertspace H. There exists a unique spectral measure on the Borel sets of, with values in B(H), suchthat

ut =(t, )d(), t g.

Proof. Let (f) =

utf(t)dt for any f L1 (G) dg. Then is a *-representation ofL1 (G)dg into

B(H). Since each (f) is a normal operator (f) f. Therefore extends by continuity to a

representation of the C algebra C0(). Restricting to the projections in the Borel functions on we obtain a spectral measure on satisfying the required relation.

Definition. Let I be an ideal in the C algebra A. We say that I is essential in A if each non-zeroclosed ideal ofA has a non-zero intersection with I.

Remark. Remember that the groups in this section are abelian. If B is a G-invariant C algebra ofA, then we can consider the dynamical system (B, G , |B). Clearly Spec(|B) Spec().

Definition. Let H(A) denote the set ofG-invariant, hereditary, non-zero C-subalgebras ofA. LetHB(A) denote the subset consisting of algebras B in H

(A) such that the closed ideal ofA generatedby B is essential in A.

The Connes spectrum of is

() =

Spec(|B) , B H

(A).

The Borchers spectrum of is

B() =

Spec(|B) , B HB(A).

Obviously () B().

Definition. If(M, G , ) is a W-dynamical system then we define

() =

Spec(|pMp) , p {non-zero G-invariant projections},

B() = Spec(|pMp) , p {non-zero G-invariant projections with c(p) = 1}.

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6.15. Theorem (Connes Subgroup). Let (A, G , ) be an abelian C-dynamical system. If 1 () and2 Spec ((), then 1 + 2 Spec(). Moreover, () is a closed subgroup of.

Proof. Let be a nbhd. of1 + 2. Then there are nbhds. 1 and 2 of 1 and 2 such that1 + 2 . Now R(2) is non-trivial by assumption; let x2 = 0 be from R(2). Let B denote

the hereditary C-subalgebra ofA generated by the orbit {t(x2x2) : t G}. Ifx B , x = 0, thent(x

2x2) = 0 for some t G. B is G-invariant so there is a non-zero element x R

|B(1). Thust(x2)x1 = 0 fro some t G.

t(x2) R(2) =t(x2)x1 R(1 + 2) =R() = 0. This holds for every a nbhd. of1 + 2, so 1 + 2 Spec().

Now, if1, 2 (), by the above construction we know 1 + 2 Spec(|B) for all B H(A).Therefore 1 + 2 (). Since () is the intersection of symmetric, closed sets, it is a closedsubgroup of.

6.16. Theorem (Z subgroups ofB). Let (A, G , ) be an abelian C-dynamical system. If B

() then n B

(), n Z.

Proof. We will prove by induction that for any nbhd. of , any B HB(A), and any n Zthere exist elements x1, . . . , xn in R

() B such that x1x2 xn = 0. This is true for n = 1 since B().

Assume the induction step for n. Let {Ci} be the maximal collection of algebras in H(B) such thatthe ideals generated by the Ci are mutually orthogonal and such that for each i there is an xi R()such that Ci is the hereditary C-algebra generated by the orbit {t(xi xi) : t G}. Let C = Ci.Either Ci HB(B) or we can find (by maximality) a closed, G-invariant idealI B, orthogonal tothe ideal generated by C such that C +I HB(B). In either case, I = 0 or I = 0, we must haveR() I = 0. Otherwise we contradict maximality of{Ci}.

C + I HB(B) and B HB(B), so C + I H(B). By the induction hypothesis there existx1, . . . , xn in R

()C +Isuch that y = x1x2 xn = 0. Since R()I = 0, xk R()C k.Thus y C. But then t(xi)y = 0 for some t G and some i since C = Ci. Since t(xi) R() B we have established the claim for n + 1, and thus for all n N.

Now assume n > 0, since B() is a symmetric set. Let n be a nbhd. ofn, and choose a nbhd.of such that + + n. Given B HB(B) we obtain x1, . . . , xn in R

() B such thaty = x1x2 xn = 0. Then y R(+ +)B R(n)B, n. Therefore n Spec(|B).But B was arbitrary.

6.2 KMS States

Remark. Now we will further specialize to the case G = R. States will be characterized by thebehaviour of their correlation functions in the complex frequency plane. Roughly speaking, the

growth at Im > 0 controls the growth for t < 0. This will introduce complex function techniques.

Definition. Let (A, G , ) be a C-dynamical system. We say that A A is analytic for if thefunction t t(A) has an extension to an analytic function (A), C.

6.17. Lemma. The set of analytic elements of a C-dynamical system for A forms a dense *-subalgebra of A. The set of analytic elements of a W-dynamical system for M forms a -weaklydense *-subalgebra ofM.

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Proof. Density follows from the approximation, for any A A,

An = 1/2n1/2

t(A)e

nt2dt,

AN

A as n , and An

is analytic. Similarly for the W case.

Definition. Given a C-dynamical system (A,R, ) we say that a state on A is a KMS state for , (0, ), if for any A Aanalytic, B A,

(B+i(A)) = ((A)B), C.

is called KMS for = 0 if it is an -invariant trace. (chaotic state) is called KMS for = if|(B(A))| A B for Im 0. (ground state)

6.18. Theorem. Let (A,R, ) be a C-dynamical system. Fix (0, ]. Then is a -KMSstate if and only if for every A, B A there exists a bounded continuous function f : C, = { C : 0 Im } , such thatf is holomorphic on int() and one of the following istrue.

If < f(t) = (Bt(A)), f(t + i) = (t(A)B).

If = f(t) = (Bt(A)), t R, f x y.

Proof. Obviously the second part implies that is -KMS. Assume is -KMS. Let {An} be thesequence of analytic elements converging to A A and let B A. Define fn() = (B(A)) for < . Then the {fn} are analytic and fn(+ i) = ((A)B).

Now each fn is bounded on , |fn()| B it(An). By the Phragmen-Lindelof theorem wehave

|fn() fm()| supz

|fn(z) fm(z)|

supt

|(Bt(An Am))| |(t(An Am)B)|

B An Am .

Therefore the {fn} are uniformly convergent to a function bounded and continuous on and holo-morphic on int(B). On the boundary f(t) = (Bt(A)), f(t + i) = (t(A)B), t R.

If = define fn() = (Bt(A)) and the KMS condition at gives |fn() fm()| An Am B for Im 0 and again the fn converge to an f with the required properties.

6.19. Theorem. Let(A,R, ) be a C-dynamical system and let be a -KMS state on A. Then is -invariant.

Proof. We have (+i(A)) = ((A)) for > 0. Thus f : ((A)) is bounded on (previous result) and periodic with period i. Therefore f is a constant. Since the analytic elementsare dense in A, is -invariant by continuity.

If = 0 the -invariance is by definition.

If = we have f : ((A)) is such that |f()| A when Im 0. Now = , so((A))

= ((A)), so for Im < 0 we have |f()| ((A)) A. Therefore f is

bounded, therefore f is a constant and is -invariant.

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6.20. Theorem (Ground States and Hamiltonians). Let (A,R, ) be a C-dynamical system andlet be a state on A. Then T.F.A.E.

1. is a KMS state with = (ground state).

2. There exists a positive operatorh on H, not necessarily bounded, with h = 0, andexp(ith)(A)exp((t(A)), t R, A A.

3. is -invariant, and if (, u, H, ) is the cyclic covariant representation associated with

then Spec

u

R+.

Proof.

2=1 Since h 0, for any A, B A we can define a function f on the upper half plane, ,holomorphic and continuous on the boundary,

f() = (exp(ih)A, B) .

Clearly |f| A B. Also, f(t) = (Bt(A)) for all t R, so satisfies the KMScondition with = .

1=3 We know that we can write ut = exp(ith) for some self-adjoint h. If A is analytic thenA is analytic for exp(ith) so f : (exp(ih)A, A) is analytic; f() = (A(A)).By assumption f() A2 if Im 0 then ((exp(h))sA, A) A

2for any s 0.

Therefore exp(h) 1, and so h 0 and Spec

u

R+.

3=2 A computation shows that with ut = expith, is in the domain ofh and h = 0.

Remark. The above theorem says something which can be readily accepted by anyone familiar with

renormalization, but only after some realignment of religious ideas. It says that the Hamiltonian

generating the time evolution depends on the state chosen, and that it does not really exist indepen-

dently. As an example, consider a spin system. In the ordered phase the Hamiltonian contains an

interaction with an external field. However, in the disordered phase this interaction is irrelevant (in

the technical sense), and the construction corresponding to the above theorem would show this. The

renormalization physics is in some way already contained inside the algebraic approach.

Definition. Let (A,R, ) be a C-dynamical system. We say that is approximately inner if thereis a net {h} A, h self-adjoint, such that

lim

(A) exp(ih)A exp(ih) = 0

uniformly on compact subsets ofC.

6.21. Theorem. Let(A,R, ) be a C-dynamical system and assume is approximately inner and1 A. Then A has a ground state (KMS state with = ).

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Proof. Let {h} be the net in the definition of approximately inner. Let t = Ad(exp(ith)).Without loss, by adding a multiple of 1 if necessary, h 0 and 0 Spec(h) for all .

Now there is a net of states {} such that (h) = 0 for each . Since the state space is compactwe can assume {} is weak-* convergent to some . |(B(A))| A B, if Im 0.

Therefore is a ground state for , applying theorem 6.18.Moreover,

|(B(A))| |( )(B(A))| + B(A) (A)+ A B

A B in the limit of.

Therefore is a ground state for .

Remark. KMS states are physically interesting because the KMS condition can be substituted for

the Gibbs ansatz, and it makes sense immediately in infinite volume, without requiring a limiting

process. To see the equivalence for finite systems, let A be a finite-dimensional matrix algebra witha canonical trace Tr (). Consider the state

(A) = Tr (A) /Tr () .

The automorphism group is t(A) = eithAeith. By elementary calculation, satisfies the KMS

condition for some , if and only if = exp(h).

6.3 Modular Group

Remark. It is a remarkable fact that von Neumann algebras carry hidden within themselves a kind ofdynamical information, in the form of an R-action. How this arises is the subject of the following.This will lead to the classification of type III factors.

Definition. Let M be a von Neumann algebra on a separable Hilbert space H. Let T be a closedoperator on H. T is said to be affiliated to M if

A Dom(T) Dom(T), T A AT, A Mc.

6.22. Lemma. LetT = U|T| be the polar decomposition of T. Then T.F.A.E.

1. U and the spectral projections E|T|() belong to M.

2. T is affiliated to M.

Proof.

Definition. Let M be a von Neumann algebra on a separable Hilbert space H. Let H be cyclicand separating for M. Define two anti-linear operators S0, F0 by

S0A A, A M,

F0B B, B Mc.

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S0 and F0 are closable. Denote their closures by Sand F respectively. Sis called the Tomita operatorfor (M, ). Furthermore we have

S0 = F F0 = S,

S

1

= S, F

1

= F,

See Ref. [BW92, p. 32].

Definition. Let S = J1/2 be the polar decomposition of the Tomita operator. The anti-unitaryoperator J is called the modular conjugation and the non-negative operator is called the modularoperator.

6.23. Lemma.

1. = F S

2. 1 = SF

3. F = J1/2

4. J = J

5. J2 = 1

6. 1/2 = J1/2J

Proof.

= SS = F S. (F S)1 = S1F1 = SF = 1.

S = S1 = 1/2J = JJ1/2J. Therefore, by uniqueness of the polar decomposition,J = J and J1/2J = J1/2J = 1/2.

J = J = J2 = 1.

Definition. The strongly continuous unitary group defined by

it

= exp(it ln).

is called the modular group.

Example. Let H = L2 ([0, 1]) dx and let M be the algebra of functions bounded a.e. on [0, 1]with pointwise multiplication. M acts on H as a commutative algebra of multiplication operators.(x) = 1 is a cyclic and separating vector for M. Then the Tomita operator is complex conjugation,SA = A, and = 1.

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Example. Let H and K be Hilbert spaces of dimension n. Let {f1, . . . , f n} and {g1, . . . , gn} beorthonormal bases for H and K respectively. Let M = B(H) C1K, acting on the total spaceH K. Define the unit vector H K by

=

nj=1

ajfj gj, aj > 0, j

|aj|2 = 1.

Note that is not an arbitrary unit vector in H K, but is diagonal in the obvious basis. is cyclicand separating for M. The action ofM on H K is generated by the following operators whichshuffle basis elements in the first factor,

Aj,s : fp gl j,pfs gl.

The Tomita operator is given by

SAj,s = Aj,s = As,j = asfj gs = S(ajfs gj),

and so

S(fs gj) =asaj

(fj gs).

From this we have

(fs gj) =

asaj

2

(fs gj),

J(fs gj) = (fj gs).

Then the spectrum of is

Spec () = Spec

1

=n

s,j=1

asaj

2.

Example. Let M(G) be the group von Neumann algebra for a locally compact group G. It is a resultthat there is a -normal and -finite weight e on M(G) such that e(xx) < if and only ifthere is a left bounded element f L2 (G) dg with (f) = x, and in this case e(xx) = f

22.

Furthermore the representation associated to e is spatially equivalent to the regular representation.See Ref. [Ped79, p. 236]. Now the unitary group associated to e is given by

(ut)(s) = it(s), L2 (G) dg, t R,

where is the modular function of the group G, which links left and right Haar measures.

Remark. The following is the fundamental result of Tomita-Takesaki theory. The most self-contained

proof is probably in [BR87, p. 94], which is what we follow. One lemma is required. A slightly dif-

ferent formalism is used in [Ped79, p. 377]. Ref. [BW92, p. 387] gives a proof in the approximately

finite dimensional (AF) case, and seems to follow [BR87] in exposition.

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6.24. Lemma. Let C, R+. LetB Mc. Then there exists an element A M such that

A = ( + 1)1B.

Furthermore we have

JBJ = 1/2A1/2 + 1/2A

1/2,

as a relation between bilinear forms on Dom(1/2) Dom(1/2).

Proof. See Ref. [BR87, p. 91-94].

6.25. Theorem (Tomita-Takesaki). LetM be a von Neumann algebra with a cyclic and separatingvector. Let,J be the associated modular operator and modular conjugation. Then

JMJ = Mc,

itMit = M, t R.

Proof. Given > 0 and B B(H), define a quadratic form

I(B) = 1/2

dtit

et + etitBit.

If, Dom(1/2) Dom(1/2), define the function

f() =

1/2, I(B)

1/2

+

1/2, I(B)

1/2

=

dtit

et + et 1/2 1/2it, B1/2it+ 1/2 1/2it, B1/2it .Let =

dE() be a spectral decomposition for . Then we have

f() =

dtit

et + et

d2(E(),BE())

1/2+

1/2

dt

et + et

it=

d2(E(),BE())

= (,B) .

Therefore, as equality of bilinear forms on Dom(1/2) Dom(1/2) we have

B = 1/2I(B)1/2 + 1/2I(B)

1/2.

From the lemma we have the existence of A1 Mc with

JA1J = 1/2A

1/2 + 1/2A1/2.

Then the above expression gives an inverse relation

A

= I

(JA1J).

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IfB Mc then, since A M,

(, [B, I(JA1J)] ) = 0,

dteipt

et + et , B, itJA1J

1/2

= 0, p R.Therefore

itJA1Jit Mcc = M.

Setting t = 0 gives JMcJ M. The symmetries of the conjugation then give JMJ Mc. UsingJ2 = 1 gives the first result. Finally, since JMcJ M, any A M has the form A = JA1J forsome A1 Mc. Since itJA1Jit M, we then have itAit M, which proves the secondclaim.

Remark. The formal calculation here is that the Fourier transform provides an inverse for the map

A JAJ M

c. The proof justifies this statement.

6.26. Theorem (Exterior Equivalence). Let and be faithful normal states on a von Neumannalgebra M. Let

Rand

Rbe the associated modular groups. Then there exists a strongly continuous

one-parameter family of unitary operators ut in M such that

1. t (x) = utt (x)u

t, x M, t R.

2. ut+s = utt (us), s, t R.

Proof. Consider the von Neumann algebra M Mat2 (C). Define a faithful normal state by

x11 x12x21 x22 = 12 ((x11) + (x22)) .Let

Rdenote the modular group We have t (x e11) = t(x) e11 and

t (y e22) = t(y) e22

for x, y M, where t and t satisfy the KMS condition for the states and . By the KMSuniqueness result then t =

t and t =

t . Define

Wt t

0 01 0

=

at ctut bt

M Mat2 (C) .

t (1) = t (1) = 1, so

Wt

Wt

= 0 00 1 , WtWt = 1 00 0 .There

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