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States on \(W^*\)-algebras and orthogonal vector measures. (English) Zbl 0743.46063

Summary: We show that every state on a \(W^*\)-algebra \({\mathcal A}\) without type \(I_ 2\) direct summand is induced by an orthogonal vector measure on \({\mathcal A}\). This result may find an application in quantum stochastics. Particularly, it allows us to find a simple formula for the transition probability between two states on \({\mathcal A}\).

MSC:

46L30 States of selfadjoint operator algebras
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
46L10 General theory of von Neumann algebras
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