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Let H be a separable Hilbert space, B(o) the bounded operators on X{, K the ideal of compact operators, and iv the natural map from 0(o{) onto the Calkin algebra $B(J)/K. Suppose X is a compact metric space and *: C(X) x [O, 1] -. (3O/X is a continuous function such that 4(, t) is a *-isomorphism for each t and such that there is a *-isomorphism +: C(X) -$(J{) with i7qj( )=Q(, 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 ': C(X) x [O, i1 (J) such that 77* = t and I(., 0)-= V/(.). Furthermore if X is the disjoint union of two spaces that both have this property, then X itself has this property. If J{ is a separable Hilbert space, Y{ is the set of compact operators on J{, and iv is the natural map from B(J{) onto the Calkin algebra %(3(J)/K then a compact metric space X is said to have the C*-covering homotopy property whenever the following holds: for any continuous map '1: C(X) x [0, 1] 0 3({),'J such that 0(t, t) is a *-isomorphism for each t and there is a *-isomorphism 0/: C(X) .53(J{) with 7O6(-) = 0(V, 0) then there is a continuous map Tl: C(X) x [0, 1] $(0{) such that T(., t) is a *-isomorphism for each t, 77T = 0, and V{)= T(I, 0). In this paper it is shown that X has the C*covering homotopy property when X is an interval, a circle, a totally disconnected space, or the disjoint union or two spaces which .have the property. Here C(X) is the C*-algebra of all continuous complex valued functions on X, and an isomorphism is understood to take the identity of the domain onto the identity of the range. The motivation to study this problem stems from the recent papers of Brown, Douglas and Fillmore [3], [4]. Indeed, if it can be shown that every compact metric space has the C-covering homotopy property, then this will Received by the editors July 8, 1974. AMS (MOS) subject classifications (1970). Primary 46L05; Secondary 55D10.

On the Calkin algebra and the covering homotopy property