When are two commutative C*-algebras stably homotopy equivalent?

Abstract

Let X and Y be finite connected CW complexes with base points, and let K denote the C*algebra of compact operators on a separable infinite dimensional complex Hilbert space. The purpose of this paper is to study the question of when C0(X)⊗K is homotopy equivalent to C0(Y )⊗K; here C0(X) is, as usual, the C*-algebra of continuous complex-valued functions which vanish at the base point. Recall that two C-algebras A and B are said to be homotopy equivalent, written A ' B, if there are ∗-homomorphisms φ : A → B and ψ : B → A for which ψ ◦ φ and φ ◦ ψ may be deformed by a path of endomorphisms to the identity maps idA : A → A and idB : B → B, respectively. Two C*-algebras A and B are called ‘stably’ homotopy equivalent if A⊗K ' B⊗K (this should not be confused with the notion of stable homotopy of spaces used in topology), so another way of stating the problem we consider is: when are C0(X) and C0(Y ) ‘stably’ homotopy equivalent? To begin with we should remark that the analogous question for homotopy equivalence has a simple answer: C0(X) and C0(Y ) are homotopy equivalent if and only if X and Y are based homotopy equivalent. The situation for ‘stable’ homotopy equivalence is more complicated and is closely related to K-theory: for example, if two C*-algebras are ‘stably’ homotopy equivalent then they have the same K-theoretic invariants. The main purpose of this paper is to show that the converse is not true: we give an example of two spaces X and Y which cannot be distinguished by