A Suspension Theorem for Continuous Trace C # -Algebras

Let B be a stable continuous trace C*-algebra with spectrum Y . We prove that the natural suspension map S, : [Co(X), B ] -+ [Co(X)@ Co(R), B @ Co(R)] is a bijection, provided that both X and Y are locally compact connected spaces whose one-point compactifications have the homotopy type of a finite CW-complex and X is noncompact. This is used to compute the second homotopy group of 9 in terms of K-theory. That is, [Co(R2), B ] = Ko(go);where Bo is a maximal ideal of B if Y is compact, and go= B if Y is noncompact. For C*-algebras A, B let Hom(A, B) denote the space of (not necessarily unital) *-homomorphisms from A to B endowed with the topology of pointwise norm convergence. By definition y, , w E Hom(A , B) are called homotopic if they lie in the same pathwise connected component of Hom(A , B) . The set of homotopy classes of the *-homomorphisms in Hom(A, B) is denoted by [A, B] . For a locally compact space X let Co(X) denote the C*-algebra of complex-valued continuous functions on X vanishing at infinity. The suspension functor S for C*-algebras is defined as follows: it takes a C*-algebra A to A €3 Co(R) and y, E Hom(A , B) to For commutative A, B this corresponds via Gelfand duality to the usual topological suspension functor. Therefore, it is natural to consider the induced map and to think about an analogue of the Freudenthal suspension theorem (see [E, R, Sc]). Let 3 denote the compact operators on an infinite separable Hilbert space. In this note we prove the following: Received by the editors April 3, 1991 and, in revised form, June 12, 1992. 1991 Mathematics Subject Classification. Primary 46L05, 46M20, 46L80; Secondary 55P40, 55P47, 19K99. The author was partially supported by NSF Grant DMS-8905875. @ 1994 American Mathematical Society