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Properly infinite AW*‐Algebras are Monotone Sequentially Complete

Properly infinite AW*‐Algebras are Monotone Sequentially Complete PROPERLY INFINITE AW-ALGEBRAS ARE MONOTONE SEQUENTIALLY COMPLETE ERIK CHRISTENSEN AND GERT K. PEDERSEN There are two outstanding problems in the theory of AW*-algebras: The existence of a trace in a finite factor, and monotone completeness of any AW*-algebra. This paper gives a partial solution of the second problem, unfortunately only in the infinite case. The finite case is now crucial in both problems. Although some operator algebraists have taken a less respectful view of AW*-algebras, we read in [2, p. 29] that "evidence has been accumulating in recent years that they will have to reconsider their position". Thus admonished by Authority, we offer the following result. THEOREM. Let A be a properly infinite AW*-algebra. Then each bounded monotone increasing sequence of self-adjoint elements has a least upper bound in A. For the proof we shall need the following five lemmas. LEMMA 1 [1, §20, Theorem 1]. If {u \ie 1} is a family of pairwise orthogonal partial isometrics in an AW*-algebra A {that is, ufUj = u,u* = Ofor i =fc j), then there is a partial isometry u in A such that u u*u = uu*u = u for all i, and { ( { *u = £ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Properly infinite AW*‐Algebras are Monotone Sequentially Complete

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/16.4.407
Publisher site
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Abstract

PROPERLY INFINITE AW-ALGEBRAS ARE MONOTONE SEQUENTIALLY COMPLETE ERIK CHRISTENSEN AND GERT K. PEDERSEN There are two outstanding problems in the theory of AW*-algebras: The existence of a trace in a finite factor, and monotone completeness of any AW*-algebra. This paper gives a partial solution of the second problem, unfortunately only in the infinite case. The finite case is now crucial in both problems. Although some operator algebraists have taken a less respectful view of AW*-algebras, we read in [2, p. 29] that "evidence has been accumulating in recent years that they will have to reconsider their position". Thus admonished by Authority, we offer the following result. THEOREM. Let A be a properly infinite AW*-algebra. Then each bounded monotone increasing sequence of self-adjoint elements has a least upper bound in A. For the proof we shall need the following five lemmas. LEMMA 1 [1, §20, Theorem 1]. If {u \ie 1} is a family of pairwise orthogonal partial isometrics in an AW*-algebra A {that is, ufUj = u,u* = Ofor i =fc j), then there is a partial isometry u in A such that u u*u = uu*u = u for all i, and { ( { *u = £

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Jul 1, 1984

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