It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer-Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer-Zadeh structure is de Morgan or, equivalently, if the range projection map preserves infima in the unit interval. This provides a new characterization of finiteness in the Murray-von Neumann structure theory of von Neumann algebras in terms of Brouwer-Zadeh structures
Cattaneo, G., Hamhalter, J. (2002). De Morgan property for effect algebras of von Neumann algebras. LETTERS IN MATHEMATICAL PHYSICS, 59(3), 243-252 [10.1023/A:1015584530597].
De Morgan property for effect algebras of von Neumann algebras
Cattaneo, G;
2002
Abstract
It is shown that the unit interval of a von Neumann algebra is a Sum Brouwer-Zadeh algebra when equipped with another unary operation sending each element to the complement of its range projection. The main result of this Letter says that a von Neumann algebra is finite if and only if the corresponding Brouwer-Zadeh structure is de Morgan or, equivalently, if the range projection map preserves infima in the unit interval. This provides a new characterization of finiteness in the Murray-von Neumann structure theory of von Neumann algebras in terms of Brouwer-Zadeh structures
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