Kakutani duality for groups Department of Mathematics University of - PowerPoint PPT Presentation

Based on a joint work V. Marra Kakutani duality for groups Department of Mathematics University of Salerno http://logica.dipmat.unisa.it/lucaspada Duality in Algebra and Logic Chapman University, 14 September 2018. Kakutani duality for groups Luca Spada The problem Motivations a-normal Luca Spada spaces

It is well known that every compact Hausdorfg space X can be embedded Embedding spaces Kakutani duality for groups Luca Spada The problem Motivations a-normal in some hypercube [0 , 1] J for some index set J. spaces Suppose that X is now endowed with a function δ : X → N . Problem Given a pair ⟨ X , δ ⟩ , is there a continuous embedding ι : X → [0 , 1] J in such a way that the denominators of the points in ι [ X ] agree with δ ? Let us assume that “agree” means that δ ( x ) = den ( ι ( x )) .

order: the top being 0 and the bottom being 1. denominator to be the natural number Denominators Kakutani duality for groups Luca Spada Recall that N forms a complete lattice under the divisibility The problem Motivations Let J be a set and p ∈ [0 , 1] J . If p ∈ Q J we defjne its a-normal spaces den ( p ) = lcd { p i | i ∈ J } where lcd stands for the least common denominator. If p ̸∈ Q J we set den ( p ) = 0 . 1. A function f : [0 , 1] J → [0 , 1] preserves denominators if for any x ∈ [0 , 1] J , den ( f ( x )) = den ( x ) . 2. A function f : [0 , 1] J → [0 , 1] respects denominators if for any x ∈ [0 , 1] J , den ( f ( x )) | den ( x ) .

in difgerent lattice points —failing continuity. lattice point —failing injectivity— or by sending the points either 0 or 1. An easy counter-example Kakutani duality for groups Luca Spada The problem Consider X = [0 , 1] with its Euclidean topology and endow it Motivations with a constant δ : a-normal spaces ∀ x ∈ X δ ( x ) = 1 . The only points with denominator equal 1 in [0 , 1] J are the so-called lattice points i.e., points whose coordinates are The only way ι could agree with δ is to send all points in one

whole variety of MV-algebras. The above mentioned problem is crucial in the duality theory this structure comes from the fact that it generates the MV-structure given by the truncated sum Łukasiewicz logic. of MV-algebras —the equivalent algebraic semantics of MV-algebras Kakutani duality for groups Luca Spada The problem Motivations An MV-algebra is a structure ⟨ A , ⊕ , ¬ , 0 ⟩ such that MV-algebras 1. ⟨ A , ⊕ , 0 ⟩ is a commutative monoid, Norm- complete 2. ¬¬ x = x , MV-algebras 3. ¬ 0 ⊕ x = ¬ 0 a-normal 4. ¬ ( ¬ x ⊕ y ) ⊕ y = ¬ ( ¬ y ⊕ x ) ⊕ x . spaces Example The interval [0 , 1] in the real numbers has a natural x ⊕ y = min { x + y , 1 } and ¬ x = 1 − x . The importance of

with integer coeffjcients, it respect denominators i.e., if it is continuous and piecewise (affjne) linear map, where Hausdorfg spaces embedded in some hypercube, with Semisimple MV-algebras with their homomorphisms form a MV-algebras and compact spaces Kakutani duality for groups Luca Spada Theorem (Marra, S. 2012) The problem Motivations category that is dually equivalent to the category of compact MV-algebras Z -maps among them. Norm- complete MV-algebras Definition a-normal For I , J arbitrary sets, a map from R I into R J is called Z -map spaces each (affjne) linear piece has integer coeffjcients. Remark Since every Z -map f acts on each point as an linear function den ( f ( x )) | den ( x ) .

preserving the order unit. follows: category of MV-algebras with their morphisms, and the and and that possesses an element u such that Mundici’s functor Kakutani duality for groups Luca Spada An abelian ℓ -group with order unit (u ℓ -group, for short), is a The problem partially ordered Abelian group G whose order is a lattice, Motivations for all g ∈ G , there exists n ∈ N such that ( n ) u ≥ g . MV-algebras Norm- The functor Γ that takes an u ℓ -group ⟨ G , + , − , 0 , u ⟩ to its complete unital interval [0 , u ] with operation ⊕ and ¬ defjned as MV-algebras a-normal spaces x ⊕ y = min { u , x + y } ¬ x = u − x , is full, faithful, and dense hence it has a quasi-inverse Ξ and Theorem (Mundici 1986) The pair Γ , Ξ gives an equivalence of categories between the category of u ℓ -groups with ordered group morphisms

which is Cauchy-complete w.r.t. its induced norm. An norm-complete MV-algebra is a semisimple MV-algebra Norm induced by the order unit Kakutani duality for groups Luca Spada The problem Definition Motivations Let ( G , u ) be a u ℓ -group. The order unit u induces a MV-algebras seminorm ∥ ∥ u defjned as folows: Norm- complete { p } ∥ g ∥ u := inf q ∈ Q | p , q ∈ N , q ̸ = 0 and q | g | ≤ pu MV-algebras a-normal The seminorm ∥ ∥ u : G → R + is in fact a norm if, and only spaces if, G is archimedean. Any semisimple MV-algebra A inherits a norm from its enveloping (archimedean) group Ξ( A ) . Definition

An answer was already given by Stone: compact Hausdorfg in the above statement? Archimedean and norm-complete (with respect to the norm Kakutani duality Kakutani duality for groups Luca Spada Theorem (Kakutani-Yosida duality 1941) The problem A unital real vector lattice ( V , u ) is isomorphic to ( C ( X ) , 1) Motivations MV-algebras for some compact Hausdorfg space X, if, and only if, V is Norm- complete ∥ ∥ u induced by the unit). MV-algebras a-normal Question spaces What if we want to substitute u ℓ -group for real vector lattice Remark spaces correspond to Archimedean, complete and divisible u ℓ -groups.

all continuous maps which respect denominators. As a corollary we obtain represented in this way. functions Denominator preserving maps Kakutani duality for groups Luca Spada Theorem (Goodearl-Handelman 1980) The problem Let X be a compact Hausdorfg space. For each x ∈ X choose Motivations A x to be either A x = R or A x = ( 1 n ) Z . Then, the algebra of MV-algebras Norm- complete MV-algebras { } f ∈ C ( X ) | f ( x ) ∈ A x for all x ∈ X , a-normal is a norm-complete u ℓ -group and every such a group can be spaces Corollary The norm-completion of the algebra of Z -maps is given by

A duality for norm-complete MV-algebras Kakutani duality for groups Luca Spada The problem The category MV Motivations Let MV be the category whose objects are semisimple MV-algebras Norm- MV-algebras and arrows are MV-homomorphisms. complete MV-algebras The category A a-normal Let A be the category whose objects are pairs ⟨ X , δ ⟩ , where spaces X is a compact Hausdorfg space and δ is a map from X into N . An arrow between two objects ⟨ X , δ ⟩ and ⟨ Y , δ ′ ⟩ is a continuous map f : X → Y that respects denominators, i.e., δ ′ ( f ( x )) | δ ( x ) .

A duality for norm-complete MV-algebras Kakutani duality for groups Luca Spada The problem Motivations The functor L MV-algebras Norm- Let L : A → MV be the assignment that associates to every complete object ⟨ X , δ ⟩ in A the MV-algebra MV-algebras a-normal L ( ⟨ X , δ ⟩ ) := { g ∈ C ( X ) | ∀ x ∈ X den ( g ( x )) | δ ( x ) } , spaces and to any A -arrow f : ⟨ X , δ ⟩ → ⟨ Y , δ ′ ⟩ the MV -arrow that sends each h ∈ L ( ⟨ Y , δ ′ ⟩ ) into the map h ◦ f .

otherwise. n A duality for norm-complete MV-algebras Kakutani duality for groups Luca Spada The problem The functor M Motivations MV-algebras Let M : MV → A be the assignment that associates to each Norm- MV-algebra A , the pair ⟨ Max ( A ) , δ A ⟩ , where Max ( A ) is complete maximal spectrum of A and, for any m ∈ Max ( A ) , MV-algebras a-normal { if A / m has n + 1 elements spaces δ A ( m ) := 0 Let also M assign to every MV-homomorphism h : A → B the map that sends every m ∈ M ( B ) into its inverse image under h , in symbols M ( h )( m ) = h − 1 [ m ] ∈ Max ( A ) .

What are the fjxed points on the topological side? side are exactly the norm-complete MV-algebras. characterise the fjxed points on each side. So, what is left to do in order to fjnd a duality is to A duality for norm-complete MV-algebras Kakutani duality for groups Luca Spada The problem Motivations Theorem MV-algebras Norm- The functors L and M form a contravariant adjunction. complete MV-algebras a-normal spaces It is quite easy to see the the fjxed points on the algebraic

A-normal spaces Kakutani duality for groups Luca Spada The problem Definition Motivations An object ⟨ X , δ ⟩ in A is said a-normal (for arithmetically normal) if for any pair of points x , y ∈ X such that x ̸ = y , a-normal 1 1. if δ ( y ) ̸ = 0 , then, letting d := spaces δ ( y ) , there exists a family of open sets { O q | q ∈ (0 , d ) ∩ Q } 2. if δ ( y ) = 0 , then there are infjnitely many d ∈ [0 , 1] such that for each of those there exists a family of open sets { O q | q ∈ (0 , d ) ∩ Q } the families { O p } are such for any p , q ∈ (0 , d ) ∩ Q and n ∈ N 1. p < q implies { x } ⊆ O p ⊆ O p ⊆ O q ⊆ O q ⊆ { y } c . 2. δ − 1 [ { n } ] ⊆ ∪ { O p | den ( p ) | n } .

a-subspaces. A-normal spaces Kakutani duality for groups Luca Spada The problem Theorem Motivations For any set I, the a-space ⟨ [0 , 1] I , den ⟩ is a-normal. a-normal spaces Lemma (A-normality is weakly hereditary) If an a-space ⟨ X , δ ⟩ is a-normal, then so are all its closed Theorem An a-space ⟨ X , δ ⟩ is a-normal if, and only if, there exist a set I and an a-iso from X into an a-subspace of ⟨ [0 , 1] I , den ⟩ .