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The ∑‐Property for Positive Linear Functionals on W ‐algebras and its Application to the Problem of Unitary Mixing in the State Space
Author(s) -
Alberti P. M.
Publication year - 1979
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19790900102
Subject(s) - mathematics , unitary state , mixing (physics) , state (computer science) , conjecture , context (archaeology) , property (philosophy) , pure mathematics , linear form , space (punctuation) , linear space , algebra over a field , discrete mathematics , mathematical analysis , quantum mechanics , law , algorithm , paleontology , philosophy , physics , epistemology , political science , biology , linguistics
In this paper we are dealing with positive linear functionals on W ‐algebras. We introduce the notion of a positive linear functional with ∑‐property (see Definition 1.1). It is shown that each positive linear functional on a W ‐algebra possesses the ∑‐property. This fact allows to give a short proof of UHLMANN's conjecture on unitary mixing in the state space of a W ‐algebra. In proving our main theorem (see Theorem 1.2.) we obtain some results on positive linear functionals and orthoprojections which are useful in other context, too.