On the Calkin Algebra and the Covering Homotopy Property

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.