Quantum Logics over Rationals

Document Type

Presentation Abstract

Presentation Date

4-1-2003

Abstract

We introduce the notion of quantum logics of idempotents in algebras of operators, and develop a measure theory for the set P(H) of all (not necessarily Hermitian) continuous linear projections on a Hilbert space H.

Any set can be considered as an idempotent in the algebra of multiplication operators, and any finitely additive measure µ can be considered as a function on these idernpotents.

Idempotents P and Q (i.e., P2 = p,Q2 = Q) in an algebra of operators are said to be orthogonal if PQ = QP = 0, in this case P + Q is also an idempotent. A function v defined on a set ∏ of idempotents is called σ-orthoadditive if v(P+Q) = v(P) =v(Q) whenever P and Q are orthogonal. v is called -orthoadditive if in addition v(ΣPn) = Σv(Pn) whenever Pn, Pm are mutually orthogonal for n≠m.

Let X be a real topological linear space and P(X) be the set of all continuous linear projections on X. For what X every relatively σ-additive function μ:P(X)→ ℝ admits an extension to a sequentially strongly continuous linear functional? Does there exist a non-Hilbert space X with this property?

Theorem: Let P(ℚn) be the set of all linear projections on ℚn, n≥3 . Then every orthoadditive function μ:P(ℚn)→ℚ defines a unique linear operator T on ℚn.

Additional Details

Tuesday, 1 April 2003
4:10 p.m. in Math 109
Coffee/treats at 3:30 p.m. Math 104 (Lounge)

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