Hamhalter, Jan Gleason property and extensions of states on projection logics. (English) Zbl 0830.46055 Bull. Lond. Math. Soc. 26, No. 4, 367-372 (1994). 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). Cited in 4 Documents 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 Keywords:state; projection logic; von Neumann algebra; Gleason property Citations:Zbl 0589.03040 PDFBibTeX XMLCite \textit{J. Hamhalter}, Bull. Lond. Math. Soc. 26, No. 4, 367--372 (1994; Zbl 0830.46055) Full Text: DOI