# 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., P ^{2} = p,Q^{2} = 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*(Σ

*P*) = Σ

_{n}*v*(

*P*

_{n}) whenever

*P*are mutually orthogonal for

_{n}, P_{m}*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*(ℚ*n*) be the set of all linear projections on ℚ^{n}, *n*≥*3* . Then every orthoadditive function *μ:P(ℚn)*→*ℚ* defines a unique linear operator *T* on *ℚ ^{n}*.

## Recommended Citation

Mushtari, Dr. Daniar H., "Quantum Logics over Rationals" (2003). *Colloquia of the Department of Mathematical Sciences*. 135.

https://scholarworks.umt.edu/mathcolloquia/135

## Additional Details

Tuesday, 1 April 2003

4:10 p.m. in Math 109

Coffee/treats at 3:30 p.m. Math 104 (Lounge)