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Gleason property and extensions of states on projection logics. (English) Zbl 0830.46055

Summary: We prove that every state on the projection logic \(P(M)\) of a von Neumann algebra \(M\) not containing a direct summand of type \(I_2\) extends to a state of an arbitrary larger unital logic \(L\). We also show that if a \(C^*\)-algebra enjoys the Gleason property, and if it possesses sufficiently many projections, then an analogous result can be derived. Moreover, we prove that the extensions can be taken linear in a complete order unit norm space associated with \(L\). (Results of this paper generalize results of P. Pták [Bull. Pol. Acad. Sci., Math. 33, 493-497 (1985; Zbl 0589.03040)] and may contribute to the noncommutative measure theory, convex theory of state spaces and foundations of quantum physics).

MSC:

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)
46L30 States of selfadjoint operator algebras
03G12 Quantum logic

Citations:

Zbl 0589.03040
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