Some Directed Subsets of C*–algebras and Semicontinuity Theory
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- by Lawrence G. Brown PDF
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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 $a\le h-\delta \mathbf {1}$ 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 $h=\mathbf {1}$, is that if $a$ and $b$ are self–adjoint in $A$, $a\le h$, and $b\le h-\delta \mathbf {1}$ for $\delta >0$, then there is a self–adjoint $c$ in $A$ such that $c\le h, a\le c$, and $b\le c$.References
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Additional Information
- Lawrence G. Brown
- Affiliation: (Emeritus) Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067
- MR Author ID: 42165
- Email: lgb@math.purdue.edu
- Received by editor(s): May 6, 2014
- Published electronically: May 1, 2015
- Communicated by: Marius Junge
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 3895-3899
- MSC (2010): Primary 46L05
- DOI: https://doi.org/10.1090/proc12744
- MathSciNet review: 3359580