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

• n Algebra Universalis
• E. Marczewski Centennial Conference, 2007

DEFINITION A measure µ on an algebra A

• f subsets of a set X is said to be continuous

if for any positive number ε there is a partition X1, . . . , Xn of X with Xi ∈ A and µ(Xi) < ε. HAMMER-SOBCZYK “Every nonnegative finitely additive measure µ on an algebra can be uniquely expressed as the sum µ0 + µi, at most denumerable, of finitely additive mea- sures such that µ0 is continuous and the µi two-valued.”

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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 (3) ∃ ! a⊥ ∈ L : a ⊥ a⊥ and a ⊕ a⊥ = 1 (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.

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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

• f 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.

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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.

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

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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.

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✉

L: effect algebra

µ: modular measure

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ✉

E: orthomodular lattice

µ: modular function

☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ✉

A: Boolean algebra

µ: measure

▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲✉ F : MV-algebra

µ: measure

☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞ ☞

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Algebraic decomposition of L Let L be a complete modular D-lattice, then L ∼ [0, b] × Πα∈A[0, α], where [0, b] is atomless and [0, α] irreducible and atomic. Topological decomposition of L If τ is an

• rder

continuous Hausdorff D- topology on L, then (L, τ) is isomorphic and homeomorphic to the product of a connected and a totally disconnected D-lattice.

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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

• perations continuous. We call its topology a

D-topology. A D-topology is order continuous if order con- vergence of nets implies topological conver- gence.

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µ continuous means that, ∀W 0-neighborhood in G and ∀a ∈ L ∃a1, . . . , an ∈ L such that a = ⊕n

i=1ai and µ([0, ai]) ⊆ 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 →

Rn be a modular measure. Then for every a ∈

L, µ([0, a]) is convex iff µ is continuous. continuous is the right condition!

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THEOREM Let G be a complete Hausdorff Abelian group and µ : L → G be an exhaustive modular measure. Then µ = λ +

• α∈A

µα where (1) λ is a continuous modular measure (2) {µα} is a uniformly summable family of modu- lar measures (3) For each α, Lα := L/N(µα) is an irreducible modular effect algebra with finite length and, for a ∈ L, µα(a) = h(ˆ a)gα, where gα ∈ G, ˆ a ∈ Lα is the equivalence class of a and h(ˆ a) is the height of ˆ a in Lα.

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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-

• nal sequence (an)n∈N in L µ(an) 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.

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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=1ai | ai

atoms such that ai 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.

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DEFINITION The diamond is the 4- 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].

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Theorem Let µ : F → G be exhaustive. Then there exist measures λ, µa (a ∈ A) such that (1) µ = λ +

a∈A µa

(2) ∀a ∈ A, F/N(µa) ∼ Lna = {0, 1

na, . . . , 1} for

some na ∈ N ⇒ µa(F) has na+1 elements, (3) λ is continuous. Decomposition of MV-algebras (Generalization of Jakubik) Let F be a complete MV-algebra. Then F ∼ F0 × [0, 1]B × Πa∈ALna where C(F0) is atomless.

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