Positive Group Homomorphisms of Free Unital Abelian -groups BLAST - - PowerPoint PPT Presentation

Positive Group Homomorphisms

• f Free Unital Abelian ℓ-groups

BLAST 2018

Tom´ aˇ s 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

1

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

1

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

2

Motivation

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

and integration

• Looking beyond Baker-Beynon duality

Finitely presented unital Abelian ℓ-groups ⇔ Rational polyhedra

2

Outline

• 1. Every G has a nonempty state space

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

3

Outline

• 1. Every G has a nonempty state space

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

• 2. A dual map St H → St G is associated with every state G → H

3

Outline

• 1. Every G has a nonempty state space

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

• 2. A dual map St H → St G is associated with every state G → H
• 3. We will explore the structure of dual maps in case that

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

3

State space

Since G is lattice ordered, its state space St G is a Bauer simplex. Define ∂ St G = {s ∈ St G | s is extremal in St G}.

4

State space

Since G is lattice ordered, its state space St G is a Bauer simplex. Define ∂ St G = {s ∈ St G | s is extremal in St G}. The strong version of Krein-Milman theorem ∂ St G is a compact Hausdorff space and St G = cl conv ∂ St G. Moreover, any s ∈ St G 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 ∂ St G. Define Val G = {v : G → R | v is a normalised ℓ-homomorphism}.

5

More about real states: extremality

The previous theorem would have a little value without a concrete description of real states in ∂ St G. Define Val G = {v : G → R | v is a normalised ℓ-homomorphism}. Theorem For every unital Abelian ℓ-group G, ∂ St G = Val G

5

More about real states: discreteness

The range s[G] of s ∈ St G 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] = 1

mZ for some m ∈ N. 6

More about real states: discreteness

The range s[G] of s ∈ St G 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] = 1

mZ for some m ∈ N.

Theorem (Goodearl) Let s ∈ St G. Then TFAE:

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

and discrete v1, . . . , vk ∈ Val G such that s =

k

󰁜

i=1

αi · vi

6

Dual map between state spaces

G St G H St H

s σs

σs(t) = t ◦ s, for all t ∈ St H

7

Dual map between state spaces

G St G H St H

s σs

σs(t) = t ◦ s, for all t ∈ St H Proposition The map σs is continuous and affine for every state s : G → H. If H is Archimedean, then s 󰀂→ σs is injective.

7

Dual map between state spaces

G St G H St H

s σs

σs(t) = t ◦ s, for all t ∈ St H 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 : St H → St G.
• Every Bauer simplex K is “free” over ∂K:

∂K K L

ι continuous τ σ ! continuous affine 8

Restricting the domain of σs

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

∂K K L

ι continuous τ σ ! continuous affine

The domain of σs is just ∂ St H = Val H G St G H Val H

s σs

σs(v) = v ◦ s, for all v ∈ Val H

8

The picture with H Archimedean

• C (Val H)

continuous functions Val H → R

• H is ℓ-isomorphic to a unital ℓ-subgroup of C (Val H)

G H Val H s(a)(v) = σs(v)(a) St G v σs(v) a s(a) s σs

9

The main question

Which continuous maps σ: Val H → St G are dual to states s : G → H ?

10

Special cases

States G → C(X) All continuous maps X → St G. Normalised ℓ-homomorphisms G → C(X) All continuous maps X → Val G.

11

Special cases

States G → C(X) All continuous maps X → St G. Normalised ℓ-homomorphisms G → C(X) All continuous maps X → Val G. 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-algebras

using Mundici’s functor:

12

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-algebras

using 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 ¯ f making the diagram commutative. S (G, u) (H, w)

ι f ¯ f ! 12

McNaughton functions

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

• continuous
• piecewise linear
• with Z coefficients

Define ∇n

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

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 → ∇n

Z. 14

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 → ∇n

Z.

• The evaluation x ∈ [0, 1]n 󰀂→ vx ∈ Val ∇n

Z is a homeomorphism

• States Fn → Fn have dual maps [0, 1]n → St ∇n

Z 14

Arithmetics of the state space

∇1

Z

∇1

Z

1 s(a)(x) = σs(x)(a) St ∇1

Z

x σs(x) a s(a) s σs

15

Arithmetics of the state space

∇1

Z

∇1

Z

1 s(a)(x) = σs(x)(a) St ∇1

Z

x σs(x) a s(a) s σs Assume that x = p

q, where p and q are coprime integers. Then σs(x) is

necessarily a discrete real state whose range is

1 mZ, where m divides q. 15

Which maps are dual to states ∇1

Z → ∇1 Z?

• We will try to find the most general maps σ inducing states

sσ : ∇1

Z → ∇1 Z, that is,

sσ(a)(x) = σ(x)(a), a ∈ ∇1

Z, x ∈ [0, 1]. 16

Which maps are dual to states ∇1

Z → ∇1 Z?

• We will try to find the most general maps σ inducing states

sσ : ∇1

Z → ∇1 Z, that is,

sσ(a)(x) = σ(x)(a), a ∈ ∇1

Z, 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 ∇1

Z

is pure if there exists f ∈ ∇1

Z whose range is in [0, 1] and

ϕ(x) = vf (x), x ∈ [x1, x2].

17

The 1st type

Definition (Pure map) Let x1, x2 ∈ [0, 1] ∩ Q and x1 < x2. A map ϕ: [x1, x2] → St ∇1

Z

is pure if there exists f ∈ ∇1

Z whose range is in [0, 1] and

ϕ(x) = vf (x), x ∈ [x1, x2].

• ϕ(x) ∈ Val ∇1

Z

• 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 = pi qi 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.

18

The 2nd type

Let x1, x2 ∈ [0, 1] ∩ Q and x1 < x2, where xi = pi qi 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 ∇1

Z is mixing if

ψ(x) = α(x) · sx1 + (1 − α(x)) · sx2, x ∈ [x1, x2], where each sxi is a discrete state with the range

1 mi Z and mi divides qi. 18

Gluing the two types together

Definition (PL state) A map σ: [0, 1] → St ∇1

Z is a PL state if there exist rationals

0 = x0 < x1 < · · · < xk = 1 and maps σi : [xi, xi+1] → St ∇1

Z

such 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 ∇1

Z be a PL state. Define

sσ(a)(x) = σ(x)(a), a ∈ ∇1

Z, x ∈ [0, 1].

Then sσ is a state ∇1

Z → ∇1 Z whose dual map is σ. 20

What next?

Conjecture The dual map of a state ∇1

Z → ∇1 Z is a PL state. 21

What next?

Conjecture The dual map of a state ∇1

Z → ∇1 Z is a PL state.

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

unital 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 ∇n

Z

valuation vx real state σ(x) rational x ∈ [0, 1]n discrete s ∈ St ∇n

Z 22