Positive Group Homomorphisms of Free Unital Abelian ℓ-groupswesfussn/slides/Kroupa.pdf · A dual map StH → StG is associated with every state G → H 3. We will explore the structure

Positive Group Homomorphisms

of Free Unital Abelian ℓ-groups

BLAST 2018

Tomáš Kroupa

The Czech Academy of Sciences

ℓ-groups and states

Unital Abelian ℓ-group (G , u)

• Group (G ,+, 0)• Lattice satisfying a ≤ b ⇒ a+ c ≤ b + c , a, b, c ∈ G• Order unit u: For every a ∈ G there is n ∈ N such that a ≤ nu

State s : (G , u) → (H,w)

• Group homomorphism• Order preserving• s(u) = w

Typically states are not ℓ-homomorphisms.

1

Motivation

• Expectation/Probability mappings are states:V. Marra – On the universal-algebraic theory of measure

and integration

• Looking beyond Baker-Beynon dualityFinitely presented unital Abelian ℓ-groups

⇔Rational polyhedra

2

Outline

1. Every G has a nonempty state space

StG = {s : G → R | s is a real state} ⊆ RG .

2. A dual map StH → StG is associated with every state G → H3. We will explore the structure of dual maps in case that

G = H = the free unital Abelian ℓ-group over 1 generator

3

Outline



2. A dual map StH → StG is associated with every state G → H

3. We will explore the structure of dual maps in case that


3

Outline



2. A dual map StH → StG is associated with every state G → H3. We will explore the structure of dual maps in case that


3

State space

Since G is lattice ordered, its state space StG is a Bauer simplex.

Define

∂ StG = {s ∈ StG | s is extremal in StG}.

The strong version of Krein-Milman theorem

∂ StG is a compact Hausdorff space and

StG = cl conv ∂ StG .

Moreover, any s ∈ StG has a unique representation (Choquet theory).

4

More about real states: extremality

The previous theorem would have a little value without a concrete

description of real states in ∂ StG .

Define

ValG = {v : G → R | v is a normalised ℓ-homomorphism}.

Theorem

For every unital Abelian ℓ-group G ,

∂ StG = ValG

5

More about real states: discreteness

The range s[G ] of s ∈ StG is an additive subgroup of R:

• s[G ] is either a dense subset of R or• an infinite cyclic subgroup of R.

In the latter case s is called discrete. Then s[G ] = 1mZ for some m ∈ N.

Theorem (Goodearl)

Let s ∈ StG . Then TFAE:

• s is discrete• There are α1, . . . ,αk ∈ [0, 1] ∩Q satisfying α1 + · · ·+ αk = 1

and discrete v1, . . . , vk ∈ ValG such that

s =k

i=1

αi · vi

6

More about real states: discreteness

The range s[G ] of s ∈ StG is an additive subgroup of R:

• s[G ] is either a dense subset of R or• an infinite cyclic subgroup of R.

In the latter case s is called discrete. Then s[G ] = 1mZ for some m ∈ N.

Theorem (Goodearl)

Let s ∈ StG . Then TFAE:

• s is discrete• There are α1, . . . ,αk ∈ [0, 1] ∩Q satisfying α1 + · · ·+ αk = 1and discrete v1, . . . , vk ∈ ValG such that

s =k

i=1

αi · vi

6

Dual map between state spaces

G StG

H StH

s σs σs(t) = t ◦ s, for all t ∈ StH

Proposition

The map σs is continuous and affine for every state s : G → H.If H is Archimedean, then s → σs is injective.

We can introduce two relevant categories

• Unital Abelian ℓ-groups with states• Bauer simplices with affine continuous maps

but adjoint functors St and Aff do not yield duality.

7

Restricting the domain of σs

• We look for a smaller representation of σs : StH → StG .• Every Bauer simplex K is “free” over ∂K :

∂K K

L

ι

continuous τ σ ! continuous affine

The domain of σs is just ∂ StH = ValH

G StG

H ValH

s σs

σs(v) = v ◦ s, for all v ∈ ValH

8

Restricting the domain of σs

• We look for a smaller representation of σs : StH → StG .• Every Bauer simplex K is “free” over ∂K :

∂K K

L

ι

continuous τ σ ! continuous affine

The domain of σs is just ∂ StH = ValH

G StG

H ValH

s σs σs(v) = v ◦ s, for all v ∈ ValH

8

The picture with H Archimedean

• C (ValH) continuous functions ValH → R• H is ℓ-isomorphic to a unital ℓ-subgroup of C (ValH)

GH

ValH0

s(a)(v) = σs(v)(a)

StG

vσs(v)

a

s(a)

s

σs

9

The main question

Which continuous maps

σ : ValH → StG

are dual to states

s : G → H

?

10

Special cases

States G → C (X )All continuous maps X → StG .

Normalised ℓ-homomorphisms G → C (X )All continuous maps X → ValG .

States Rn → Rn

Stochastic matrices of order n.

States Zn → Zn

Stochastic matrices of order n with {0, 1} entries.

11

Free unital Abelian ℓ-groups

• Unital Abelian ℓ-groups do not form a variety of algebras.• However, we can rephrase the universal property of free MV-algebrasusing Mundici’s functor:

Definition

A unital Abelian ℓ-group (G , u) is free over S if there is a function

ι : S → (G , u) with ι[S ] ⊆ [0, u] and such that for any f : S → (H,w)with f [S ] ⊆ [0,w ], there is a unique normalised ℓ-homomorphism f̄making the diagram commutative.

S (G , u)

(H,w)

ι

f f̄ !

12

McNaughton functions

A McNaughton function is a function a : [0, 1]n → R that is

• continuous• piecewise linear• with Z coefficients

Define

∇nZ = {a : [0, 1]n → R | a is a McNaughton function}

0 1

13

Representation of free unital Abelian ℓ-groups

Unital version of Baker-Beynon theorem

Let (Fn, u) be the free unital Abelian ℓ-group over {g1, . . . , gn} ⊆ [0, u]and let πi : [0, 1]

n → R be the i-th coordinate projection.The map gi → πi extends uniquely to a normalised ℓ-isomorphism

Fn → ∇nZ.

• The evaluation x ∈ [0, 1]n → vx ∈ Val∇nZ is a homeomorphism• States Fn → Fn have dual maps [0, 1]n → St∇nZ

14

Arithmetics of the state space

∇1Z∇1Z

10

s(a)(x) = σs(x)(a)

St∇1Z

xσs(x)

a

s(a)

s

σs

Assume that x = pq , where p and q are coprime integers. Then σs(x) is

necessarily a discrete real state whose range is 1mZ, where m divides q.

15

Arithmetics of the state space

∇1Z∇1Z

10

s(a)(x) = σs(x)(a)

St∇1Z

xσs(x)

a

s(a)

s

σs

Assume that x = pq , where p and q are coprime integers. Then σs(x) is

necessarily a discrete real state whose range is 1mZ, where m divides q.15

Which maps are dual to states ∇1Z → ∇1Z?

• We will try to find the most general maps σ inducing statessσ : ∇1Z → ∇1Z, that is,

sσ(a)(x) = σ(x)(a), a ∈ ∇1Z, x ∈ [0, 1].

• We identify two types of such maps and then glue them together.

16

The 1st type

Definition (Pure map)

Let x1, x2 ∈ [0, 1] ∩Q and x1 < x2. A map

ϕ : [x1, x2] → St∇1Z

is pure if there exists f ∈ ∇1Z whose range is in [0, 1] and

ϕ(x) = vf (x), x ∈ [x1, x2].

• ϕ(x) ∈ Val∇1Z• If ϕ is pure on [0, 1], then sϕ is a normalised ℓ-homomorphism

17

The 2nd type

Let x1, x2 ∈ [0, 1] ∩Q and x1 < x2, where

xi =piqi

coprime pi ≥ 0, qi > 0.

and p2q1 − p1q2 = 1.

There is a unique linear function with Z coefficients α : [x1, x2] → [0, 1]such that α(x1) = 1 and α(x2) = 0.

Definition (Mixing map)

A map ψ : [x1, x2] → St∇1Z is mixing if

ψ(x) = α(x) · sx1 + (1− α(x)) · sx2 , x ∈ [x1, x2],

where each sxi is a discrete state with the range1miZ and mi divides qi .

18

Gluing the two types together

Definition (PL state)

A map σ : [0, 1] → St∇1Z is a PL state if there exist rationals

0 = x0 < x1 < · · · < xk = 1

and maps

σi : [xi , xi+1] → St∇1Zsuch that each σi is either pure or mixing, and

σ(x) = σi (x), x ∈ [xi , xi+1], i = 0, . . . , k − 1.

19

Every PL state induces a state

Theorem

Let σ : [0, 1] → St∇1Z be a PL state. Define

sσ(a)(x) = σ(x)(a), a ∈ ∇1Z, x ∈ [0, 1].

Then sσ is a state ∇1Z → ∇1Z whose dual map is σ.

20

What next?

Conjecture

The dual map of a state ∇1Z → ∇1Z is a PL state.

• Disprove/prove the conjecture.• If possible, extend the results to states between finitely presented

unital Abelian ℓ-groups.

21

What next?

Conjecture

The dual map of a state ∇1Z → ∇1Z is a PL state.

• Disprove/prove the conjecture.• If possible, extend the results to states between finitely presentedunital Abelian ℓ-groups.

21

Wishlist

normalised ℓ-homomorphism state

PL map with Z coefficients PL state[0, 1]m → [0, 1]n [0, 1]m → St∇nZvaluation vx real state σ(x)

rational x ∈ [0, 1]n discrete s ∈ St∇nZ

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Documents

Positive Group Homomorphisms of Free Unital Abelian ℓ-groupswesfussn/slides/Kroupa.pdf · A dual map StH → StG is associated with every state G → H 3. We will explore the structure