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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
Published electronically: May 1, 2015
MathSciNet review: 3359580
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Abstract: The main result concerns a $ \sigma -$unital $ C^*$-algebra $ A$, a strongly lower semicontinuous element $ h$ of $ A^{**}$, the enveloping von Neumann algebra, and the set of self-adjoint elements $ a$ of $ A$ such that $ a\le h-\delta \b 1$ for some $ \delta >0$, where 1 is the identity of $ A^{**}$. The theorem is that this set is directed upward. It follows that if this set is non-empty, then $ h$ is the limit of an increasing net of self-adjoint elements of $ A$. A complement to the main result, which may be new even if $ h=\b 1$, is that if $ a$ and $ b$ are self-adjoint in $ A$, $ a\le h$, and $ b\le h-\delta \b 1$ for $ \delta >0$, then there is a self-adjoint $ c$ in $ A$ such that $ c\le h, a\le c$, and $ b\le c$.

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

Received by editor(s): May 6, 2014
Published electronically: May 1, 2015
Communicated by: Marius Junge
Article copyright: © Copyright 2015 American Mathematical Society

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