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
.
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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