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A Necessary and Sufficient Condition for a Von Neumann Algebra to be in Standard Form
Author(s) -
Rousseau Ronny,
van Daele Alfons,
Vanheeswijck Lutgarde
Publication year - 1977
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-15.1.147
Subject(s) - von neumann architecture , von neumann algebra , mathematics , involution (esoterism) , generalization , pure mathematics , vector space , abelian von neumann algebra , algebra over a field , hilbert space , conjugate , jordan algebra , mathematical analysis , algebra representation , politics , political science , law
If M is a von Neumann algebra acting on a Hilbert space H , then M is said to be in standard form if there is a conjugate linear involution J on H such that JMJ = M ′ and JaJ = a * for all a ∈ M ∩ M ′. It is proved that M is in standard form if and only if there exists a family {ξ α } in H which is cyclic both for M and M ′ and such that the spaces { M ξ α } are mutually orthogonal, as well as the spaces { M ′ξ α }. This is a generalization of the corresponding result for σ‐finite von Neumann algebras in which case the family {ξ α } is replaced by a single vector ξ.