On the Calkin algebra and the covering homotopy property

Abstract

Let H \mathcal {H} be a separable Hilbert space, B ( H ) \mathcal {B}(\mathcal {H}) the bounded operators on H , K \mathcal {H},\mathcal {K} the ideal of compact operators, and π \pi the natural map from B ( H ) \mathcal {B}(\mathcal {H}) onto the Calkin algebra B ( H ) / K \mathcal {B}(\mathcal {H})/\mathcal {K} . Suppose X is a compact metric space and Φ : C ( X ) × [ 0 , 1 ] → B ( H ) / K \Phi :C(X) \times [0,1] \to \mathcal {B}(\mathcal {H})/\mathcal {K} is a continuous function such that Φ ( ⋅ , t ) \Phi ( \cdot ,t) is a ∗ \ast -isomorphism for each t and such that there is a ∗ \ast -isomorphism ψ : C ( X ) → B ( H ) \psi :C(X) \to \mathcal {B}(\mathcal {H}) with π ψ ( ⋅ ) = Φ ( ⋅ , 0 ) \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 Ψ : C ( X ) × [ 0 , 1 ] → B ( H ) \Psi :C(X) \times [0,1] \to \mathcal {B}(\mathcal {H}) such that π Ψ = Φ \pi \Psi = \Phi and Ψ ( ⋅ , 0 ) = ψ ( ⋅ ) \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.