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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On the Calkin algebra and the covering homotopy property


Author: John B. Conway
Journal: Trans. Amer. Math. Soc. 211 (1975), 135-142
MSC: Primary 46L05; Secondary 46M20, 55F05, 58G10
MathSciNet review: 0399875
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Abstract: Let $ \mathcal{H}$ be a separable Hilbert space, $ \mathcal{B}(\mathcal{H})$ the bounded operators on $ \mathcal{H},\mathcal{K}$ the ideal of compact operators, and $ \pi $ the natural map from $ \mathcal{B}(\mathcal{H})$ onto the Calkin algebra $ \mathcal{B}(\mathcal{H})/\mathcal{K}$. Suppose X is a compact metric space and $ \Phi :C(X) \times [0,1] \to \mathcal{B}(\mathcal{H})/\mathcal{K}$ is a continuous function such that $ \Phi ( \cdot ,t)$ is a $ \ast$-isomorphism for each t and such that there is a $ \ast$-isomorphism $ \psi :C(X) \to \mathcal{B}(\mathcal{H})$ with $ \pi \psi ( \cdot ) = \Phi ( \cdot ,0)$. It is shown in this paper that if X is a simple Jordan curve, a simple closed Jordan curve, or a totally disconnected metric space then there is a continuous map $ \Psi :C(X) \times [0,1] \to \mathcal{B}(\mathcal{H})$ such that $ \pi \Psi = \Phi $ and $ \Psi ( \cdot ,0) = \psi ( \cdot )$. Furthermore if X is the disjoint union of two spaces that both have this property, then X itself has this property.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0399875-4
PII: S 0002-9947(1975)0399875-4
Keywords: Operators on a Hilbert space, Calkin algebra, covering homotopy property
Article copyright: © Copyright 1975 American Mathematical Society