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

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)
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