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Homotopy Classification of Projections in the Corona Algebra of a Non-simple C*-algebra

Published online by Cambridge University Press:  20 November 2018

Lawrence G. Brown
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, USA 47907 email: lgb@math.purdue.edu
Hyun Ho Lee
Affiliation:
Department of Mathematics, University of Ulsan, Ulsan, Korea 680-749 email: hadamard@ulsan.ac.kr
Corresponding
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Abstract

We study projections in the corona algebra of $C\left( X \right)\,\otimes \,K$ , where $K$ is the ${{C}^{*}}$ -algebra of compact operators on a separable infinite dimensional Hilbert space and $X\,=\,[0,\,1],\,[0,\,\infty ),\,(-\infty ,\,\infty ),\,\text{or}\,\text{ }\!\![\!\!\text{ 0,}\,\text{1 }\!\!]\!\!\text{ / }\!\!\{\!\!\text{ 0,}\,\text{1 }\!\!\}\!\!\text{ }$ . Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in ${{K}_{0}}$ , Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2012

References

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