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 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 orthoadditive if whenever P and Q are orthogonal. v is called -orthoadditive if in addition whenever are mutually orthogonal for

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 -additive function admits an extension to a sequentially strongly continuous linear functional? Does there exist a non-Hilbert space X with this property?

Theorem: Let be the set of all linear projections on . Then every orthoadditive function defines a unique linear operator T on

Spring 2003 Colloquium Schedule | Mathematical Sciences | The University of Montana