Some Directed Subsets of C*-algebras and Semicontinuity Theory

Author:
Lawrence G. Brown

Journal:
Proc. Amer. Math. Soc. **143** (2015), 3895-3899

MSC (2010):
Primary 46L05

DOI:
https://doi.org/10.1090/proc12744

Published electronically:
May 1, 2015

MathSciNet review:
3359580

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Abstract | References | Similar Articles | Additional Information

Abstract: The main result concerns a unital -algebra , a strongly lower semicontinuous element of , the enveloping von Neumann algebra, and the set of self-adjoint elements of such that for some , where **1** is the identity of . The theorem is that this set is directed upward. It follows that if this set is non-empty, then is the limit of an increasing net of self-adjoint elements of . A complement to the main result, which may be new even if , is that if and are self-adjoint in , , and for , then there is a self-adjoint in such that , and .

**[1]**Charles A. Akemann and Gert K. Pedersen,*Complications of semicontinuity in -algebra theory*, Duke Math. J.**40**(1973), 785-795. MR**0358361 (50 #10827)****[2]**Lawrence G. Brown,*Semicontinuity and multipliers of -algebras*, Canad. J. Math.**40**(1988), no. 4, 865-988. MR**969204 (90a:46148)**, https://doi.org/10.4153/CJM-1988-038-5**[3]**Edward G. Effros,*Order ideals in a -algebra and its dual*, Duke Math. J.**30**(1963), 391-411. MR**0151864 (27 #1847)****[4]**Gert K. Pedersen,*Applications of weak semicontinuity in -algebra theory*, Duke Math. J.**39**(1972), 431-450. MR**0315463 (47 #4012)****[5]**Gert K. Pedersen,*-algebras and their automorphism groups*, London Mathematical Society Monographs, vol. 14, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1979. MR**548006 (81e:46037)**

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Additional Information

**Lawrence G. Brown**

Affiliation:
(Emeritus) Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067

Email:
lgb@math.purdue.edu

DOI:
https://doi.org/10.1090/proc12744

Received by editor(s):
May 6, 2014

Published electronically:
May 1, 2015

Communicated by:
Marius Junge

Article copyright:
© Copyright 2015
American Mathematical Society