On the Lebesgue decomposition of the normal states of a JBW-algebra. (English) Zbl 0762.46061

Dubois, Jacques; Hadjou, Brahim

Math. Bohem. 117, No. 2, 185-193 (1992).

Summary: A theorem is proved asserting that any linear functional defined on a JBW-algebra admits a Lebesgue decomposition with respect to any normal state defined on the algebra. Then we show that the positivity (and the unicity) of this decomposition is insured for the trace states defined on the algebra. In fact, this property can be used to give a new characterization of the trace states amongst all the normal states.

MSC:

46L30	States of selfadjoint operator algebras
46L51	Noncommutative measure and integration
46L53	Noncommutative probability and statistics
46L54	Free probability and free operator algebras
46L70	Nonassociative selfadjoint operator algebras
06C15	Complemented lattices, orthocomplemented lattices and posets
46H70	Nonassociative topological algebras
81P10	Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)

Keywords:

linear functional; JBW-algebra; Lebesgue decomposition; normal state; trace states