TY - JOUR
AU - Pedersen, Gert K.
AB - 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 = £ 
TI - Properly infinite AW*‐Algebras are Monotone Sequentially Complete
JO - Bulletin of the London Mathematical Society
DO - 10.1112/blms/16.4.407
DA - 1984-07-01
UR - https://www.deepdyve.com/lp/wiley/properly-infinite-aw-algebras-are-monotone-sequentially-complete-rMfuNCbj4x
SP - 407
EP - 410
VL - 16
IS - 4
DP - DeepDyve
ER -