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

Full-text PDF

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

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
46L05

Retrieve articles in all journals with MSC (2010): 46L05

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