Decomposition of effect algebras and the Hammer-Sobczyk theorem A - PDF document

Decomposition of effect algebras and the Hammer-Sobczyk theorem A report of the joint paper by Anna Avallone, Giuseppina Barbieri Paolo Vitolo and Hans Weber on Algebra Universalis E. Marczewski Centennial Conference, 2007

DEFINITION A measure µ on an algebra A of subsets of a set X is said to be continuous if for any positive number ε there is a partition X 1 , . . . , X n of X with X i ∈ A and µ ( X i ) < ε . HAMMER-SOBCZYK “ Every nonnegative finitely additive measure µ on an algebra can be uniquely expressed as the sum µ 0 + � µ i , at most denumerable, of finitely additive mea- sures such that µ 0 is continuous and the µ i two-valued. ” 1

EFFECT ALGEBRA ( L, ⊕ , 0 , 1) ⊕ : a partially defined operation where a ⊥ b means that a ⊕ b is defined (1) If a ⊥ b then b ⊥ a and a ⊕ b = b ⊕ a (2) If b ⊥ c and a ⊥ ( b ⊕ c ) then a ⊥ b, ( a ⊕ b ) ⊥ c and a ⊕ ( b ⊕ c ) = ( a ⊕ b ) ⊕ c ∃ ! a ⊥ ∈ L : a ⊥ a ⊥ a ⊕ a ⊥ = 1 (3) and (4) If a ⊥ 1 then a = 0 ⊖ : another partially defined operation c ⊖ a exists and equals b � a ⊕ b exists and equals c In particular a ⊥ = 1 ⊖ a . 2

The sum of a finite sequence is defined by in- duction and is independent on permutations. A finite sequence is said to be “orthogonal” if its sum is defined. An infinite sequence is orthogonal if all its par- tial sums are defined. In this case we define the sum of the infinite sequence as the supremum of its partial sums (provided that it exists). ≤ Ordering relation : a ≤ c � ∃ b ∈ L : a ⊕ b exists and equals c Hence c ⊖ a is defined if and only if a ≤ c . Moreover a ⊥ b if and only if a ≤ b ⊥ . A lattice-ordered effect algebra is also called D-lattice . 3

Effect algebras have been introduced in 1994 by Foulis and Bennett, in order to construct models of quantum-mechanical systems with unsharp measurements. Examples of effect algebras (1) An orthomodular poset can be viewed as an effect algebra by setting a ⊥ b iff a ≤ b ⊥ and a ⊕ b = a ∨ b (2) An MV-algebra can be viewed as an effect al- gebra by setting a ⊥ b iff a ≤ b ′ and a ⊕ b = a + b . 4

DEFINITION An orthomodular poset ( E, ⊥ , ≤ ) is a poset ( E, ≤ ) endowed with a unary operation ⊥ , namely the orthocomple- mentation, which satisfies x ⊥ ∨ x = 1 and x ⊥ ∧ x = 0 (1) ( x ⊥ ) ⊥ = x (2) x ≤ y ⇒ y = x ∨ ( y ∧ x ⊥ ) for every x, y ∈ E. (3) DEFINITION An MV-algebra ( F, + , ′ ; 0 , 1) is a commutative semigroup ( F, +) and ′ : F → F satisfies : (1) x + 1 = 1 ( x ′ ) ′ = x (2) 0 ′ = 1 (3) ( x ′ + y ) ′ + y = ( x + y ′ ) ′ + x for every x, y ∈ F. (4) 5

If G is a group, a measure on an effect algebra L is a map µ : L → G such that a ⊥ b ⇒ µ ( a ⊕ b ) = µ ( a ) + µ ( b ) . If L is a D-lattice, we say that µ : L → G is a modular function if µ ( a ) + µ ( b ) = µ ( a ∨ b ) + µ ( a ∧ b ) . If µ : L → G is a function and L is a D-lattice, then µ is a modular measure iff, for every a, b ∈ L, µ ( a ) = µ (( a ∨ b ) ⊖ b ) + µ ( a ∧ b ) . In particular, every measure on an MV-algebra is modular, and every modular function µ on an orthomodular lattice, with µ (0) = 0, is a measure. In this talk we present a result concerning mod- ular measures on a D-lattice L with values in a Hausdorff topological Abelian group G. 6

A : Boolean algebra µ : measure ✉ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ▲ ✉ F : MV-algebra ☞ ▲ E : orthomodular lattice ✉ ▲ ☞ ▲ µ : measure µ : modular function ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ▲ ☞ ▲ ☞ ▲ ☞ ▲ ☞ ☞ ✉ L : effect algebra µ : modular measure 7

Algebraic decomposition of L Let L be a complete modular D -lattice, then L ∼ [0 , b ] × Π α ∈ A [0 , α ] , where [0 , b ] is atomless and [0 , α ] irreducible and atomic. Topological decomposition of L If is an order continuous Hausdorff D - τ topology on L , then ( L, τ ) is isomorphic and homeomorphic to the product of a connected and a totally disconnected D -lattice. 8

An element b covers a if [ a, b ] = { a, b } . An element which covers 0 is an atom . L is atomless if it doesn’t contain any atoms. L is atomic if ∀ b � = 0 ∃ a atom a ≤ b . A topological D -lattice is a D -lattice en- dowed with a topology which makes all the operations continuous. We call its topology a D -topology . A D -topology is order continuous if order con- vergence of nets implies topological conver- gence. 9

µ continuous means that, ∀ W 0-neighborhood in G and ∀ a ∈ L ∃ a 1 , . . . , a n ∈ L such that a = ⊕ n i =1 a i and µ ([0 , a i ]) ⊆ W . Every continuous modular measure is atom- less , i.e. for every a ∈ L with µ ( a ) � = 0, there exists b < a such that µ ( b ) � = 0 and µ ( b ) � = µ ( a ). If L is σ -complete and µ is σ -additive, then µ is continuous if and only if is atomless. THEOREM Let L be σ -complete and µ : L → R n be a modular measure. Then for every a ∈ L, µ ([0 , a ]) is convex iff µ is continuous. continuous is the right condition! 10

THEOREM Let G be a complete Hausdorff Abelian group and µ : L → G be an exhaustive modular measure. Then � µ = λ + µ α α ∈ A where (1) λ is a continuous modular measure (2) { µ α } is a uniformly summable family of modu- lar measures (3) For each α, L α := L/N ( µ α ) is an irreducible modular effect algebra with finite length and, for a ∈ L , µ α ( a ) = h (ˆ a ) g α , where g α ∈ G , ˆ a ∈ L α is the equivalence class of a and h (ˆ a ) is the height of ˆ a in L α . 11

The height of L is the least upper bound of lengths of finite chains of L (the length of a finite chain with n elements is n − 1). If L has finite height, the height of an element a ∈ L is the least upper bound of lengths of the chains between a and 0. A measure µ is exhaustive if for every orthog- onal sequence ( a n ) n ∈ N in L µ ( a n ) converges to 0. An element p of L is central if for every a ∈ L ( a ∧ p ) ∨ ( a ∧ p ⊥ ) = a. The center C ( L ) of L is the set of all central elements. We stress that it is a Boolean sub- algebra of the center in the lattice theoretical sense. 12

A tool THEOREM In a complete modular atomic irreducible D -lattice different from a diamond, any two atoms are relatively perspective. Sketch of the proof Fix an atom e and define S = {⊕ n i =1 a i | a i atoms such that a i and e are relatively per- spective } Then prove (1) S is a D -ideal (2) S coincides with the set of the atoms. COROLLARY Any two atoms have the same measure. 13

The is the 4- DEFINITION diamond elements D -lattice consisting of 0 , 1 and two different atoms a, b such that a ⊥ = a and b ⊥ = b . DEFINITION A D -ideal is a non-empty sub- set I closed under sum and which satisfies the following rule a ∈ I ⇒ ( a ∨ c ) ⊖ c ∈ I ∀ c ∈ L. DEFINITION Two elements are relatively perspective if for some t ∈ L they have a com- mon relative complement in [0 , t ]. 14

Theorem Let µ : F → G be exhaustive. Then there exist measures λ, µ a ( a ∈ A ) such that (1) µ = λ + � a ∈ A µ a ∀ a ∈ A , F/N ( µ a ) ∼ L n a = { 0 , 1 (2) n a , . . . , 1 } for some n a ∈ N ⇒ µ a ( F ) has n a +1 elements, (3) λ is continuous. Decomposition of MV-algebras (Generalization of Jakubik) Let F be a complete MV-algebra. Then F ∼ F 0 × [0 , 1] B × Π a ∈ A L n a where C ( F 0 ) is atomless. 15