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


References [Enhancements On Off] (What's this?)

  • [1] Charles A. Akemann and Gert K. Pedersen, Complications of semicontinuity in $ C^{\ast } $-algebra theory, Duke Math. J. 40 (1973), 785-795. MR 0358361 (50 #10827)
  • [2] Lawrence G. Brown, Semicontinuity and multipliers of $ C^*$-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 $ C^{\ast } $-algebra and its dual, Duke Math. J. 30 (1963), 391-411. MR 0151864 (27 #1847)
  • [4] Gert K. Pedersen, Applications of weak$ ^{\ast } $ semicontinuity in $ C^{\ast } $-algebra theory, Duke Math. J. 39 (1972), 431-450. MR 0315463 (47 #4012)
  • [5] Gert K. Pedersen, $ C^{\ast } $-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

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