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

Positive Group Homomorphisms of 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 For every a ∈ G there is n ∈ N such that a ≤ nu • Order unit u : 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 For every a ∈ G there is n ∈ N such that a ≤ nu • Order unit u : 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 For every a ∈ G there is n ∈ N such that a ≤ nu • Order unit u : 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 } ⊆ R G . 3

Outline 1. Every G has a nonempty state space St G = { s : G → R | s is a real state } ⊆ R G . 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 } ⊆ R G . 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 Hausdor ff 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 m Z 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 m Z for some m ∈ N . Theorem (Goodearl) Let s ∈ St G . Then TFAE: • s is discrete • There are α 1 , . . . , α k ∈ [0 , 1] ∩ Q satisfying α 1 + · · · + α k = 1 and discrete v 1 , . . . , v k ∈ Val G such that k 󰁜 s = α i · v i i =1 6

Dual map between state spaces G St G σ s ( t ) = t ◦ s , for all t ∈ St H σ s s H St H 7

Dual map between state spaces G St G σ s ( t ) = t ◦ s , for all t ∈ St H σ s s H St H Proposition The map σ s is continuous and a ffi ne for every state s : G → H . If H is Archimedean, then s 󰀂→ σ s is injective. 7

Dual map between state spaces G St G σ s ( t ) = t ◦ s , for all t ∈ St H σ s s H St H Proposition The map σ s is continuous and a ffi ne for every state s : G → H . If H is Archimedean, then s 󰀂→ σ s is injective. We can introduce two relevant categories • Unital Abelian ℓ -groups with states • Bauer simplices with a ffi ne continuous maps but adjoint functors St and A ff do not yield duality. 7

Restricting the domain of σ s • We look for a smaller representation of σ s : St H → St G . • Every Bauer simplex K is “free” over ∂ K : ι ∂ K K continuous τ σ ! continuous a ffi ne L 8

Restricting the domain of σ s • We look for a smaller representation of σ s : St H → St G . • Every Bauer simplex K is “free” over ∂ K : ι ∂ K K continuous τ σ ! continuous a ffi ne L The domain of σ s is just ∂ St H = Val H G St G σ s ( v ) = v ◦ s , for all v ∈ Val H σ s s H Val H 8

The picture with H Archimedean continuous functions Val H → R • C (Val H ) • H is ℓ -isomorphic to a unital ℓ -subgroup of C (Val H ) s H G s ( a ) a s ( a )( v ) = σ s ( v )( a ) St G σ s ( v ) 0 Val H v σ s 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 R n → R n Stochastic matrices of order n . States Z n → Z n 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. ι S ( G , u ) f ¯ f ! ( H , w ) 12

McNaughton functions A McNaughton function is a function a : [0 , 1] n → R that is • continuous • piecewise linear • with Z coe ffi cients Define Z = { a : [0 , 1] n → R | a is a McNaughton function } ∇ n 0 1 13

Representation of free unital Abelian ℓ -groups Unital version of Baker-Beynon theorem Let ( F n , u ) be the free unital Abelian ℓ -group over { g 1 , . . . , g n } ⊆ [0 , u ] and let π i : [0 , 1] n → R be the i -th coordinate projection. The map g i 󰀂→ π i extends uniquely to a normalised ℓ -isomorphism F n → ∇ n Z . 14

Representation of free unital Abelian ℓ -groups Unital version of Baker-Beynon theorem Let ( F n , u ) be the free unital Abelian ℓ -group over { g 1 , . . . , g n } ⊆ [0 , u ] and let π i : [0 , 1] n → R be the i -th coordinate projection. The map g i 󰀂→ π i extends uniquely to a normalised ℓ -isomorphism F n → ∇ n Z . • The evaluation x ∈ [0 , 1] n 󰀂→ v x ∈ Val ∇ n Z is a homeomorphism • States F n → F n have dual maps [0 , 1] n → St ∇ n Z 14

Arithmetics of the state space s ∇ 1 Z ∇ 1 Z s ( a ) a s ( a )( x ) = σ s ( x )( a ) St ∇ 1 Z σ s ( x ) 0 1 x σ s 15

Arithmetics of the state space s ∇ 1 Z ∇ 1 Z s ( a ) a s ( a )( x ) = σ s ( x )( a ) St ∇ 1 Z σ s ( x ) 0 1 x σ s Assume that x = p q , where p and q are coprime integers. Then σ s ( x ) is m Z , where m divides q . 1 necessarily a discrete real state whose range is 15

Which maps are dual to states ∇ 1 Z → ∇ 1 Z ? • We will try to find the most general maps σ inducing states s σ : ∇ 1 Z → ∇ 1 Z , that is, a ∈ ∇ 1 s σ ( a )( x ) = σ ( x )( a ) , 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 s σ : ∇ 1 Z → ∇ 1 Z , that is, a ∈ ∇ 1 s σ ( a )( x ) = σ ( x )( a ) , Z , x ∈ [0 , 1] . • We identify two types of such maps and then glue them together. 16

The 1st type Definition (Pure map) Let x 1 , x 2 ∈ [0 , 1] ∩ Q and x 1 < x 2 . A map ϕ : [ x 1 , x 2 ] → St ∇ 1 Z is pure if there exists f ∈ ∇ 1 Z whose range is in [0 , 1] and ϕ ( x ) = v f ( x ) , x ∈ [ x 1 , x 2 ] . 17

The 1st type Definition (Pure map) Let x 1 , x 2 ∈ [0 , 1] ∩ Q and x 1 < x 2 . A map ϕ : [ x 1 , x 2 ] → St ∇ 1 Z is pure if there exists f ∈ ∇ 1 Z whose range is in [0 , 1] and ϕ ( x ) = v f ( x ) , x ∈ [ x 1 , x 2 ] . • ϕ ( x ) ∈ Val ∇ 1 Z • If ϕ is pure on [0 , 1], then s ϕ is a normalised ℓ -homomorphism 17

The 2nd type Let x 1 , x 2 ∈ [0 , 1] ∩ Q and x 1 < x 2 , where x i = p i coprime p i ≥ 0 , q i > 0 . q i and p 2 q 1 − p 1 q 2 = 1. There is a unique linear function with Z coe ffi cients α : [ x 1 , x 2 ] → [0 , 1] such that α ( x 1 ) = 1 and α ( x 2 ) = 0. 18