On the Calkin algebra and the covering homotopy property
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- by John B. Conway PDF
- Trans. Amer. Math. Soc. 211 (1975), 135-142 Request permission
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.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 211 (1975), 135-142
- MSC: Primary 46L05; Secondary 46M20, 55F05, 58G10
- DOI: https://doi.org/10.1090/S0002-9947-1975-0399875-4
- MathSciNet review: 0399875