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Universal state space embeddability of Jordan-Banach algebras. (English) Zbl 0907.46052

Summary: We study extensions of states between projection structures of JB algebras and generalized orthomodular posets. It is shown that projection orthoposet of a JB algebra \(A\) admits the universal extension property if and only if the Gleason theorem is valid for \(A\). As a consequence we get that any positive Stone algebra-valued measure on the projection lattice of a quotient of a JBW algebra without type \(I_2\) direct summand extends to a positive measure on an arbitrary larger generalized orthomodular lattice.

MSC:

46L70 Nonassociative selfadjoint operator algebras
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
28B15 Set functions, measures and integrals with values in ordered spaces
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
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