The Cuntz semigroup as an invariant for C*-algebras

A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits. 1. Recently, Toms in [26] used the refinement of the invariant K0 introduced by Cuntz almost thirty years ago in [4] to show that certain C*-algebras are not isomorphic. Anticipating the possible use of this invariant to establish isomorphism, we take the liberty of reporting some observations concerning it. (In particular, we present what might be viewed as an embryonic isomorphism theorem.) One of the first things that might be noted in connection with this invariant, which considers, instead of the finitely generated projective modules over a given C*-algebra, the larger class of modules consisting of the countably generated Hilbert C*-modules (see [12], [15], and [21]; see also [10]) is that, whereas the equivalence relation between finitely generated projective modules would appear to be inevitable, namely, just isomorphism, in the wider setting of Hilbert C*-modules it is no longer quite so clear what the equivalence relation should be. While it is tempting just to choose isomorphism again, one should note that, even in the stably finite case (which is perhaps the case that this invariant is of most interest), whereas the isomorphism classes of algebraically finitely generated Hilbert C*-modules (which are of course also algebraically projective, and up to isomorphism exhaust the finitely generated projective modules) form an ordered set with respect to inclusion (in other words, if each of two such modules is isomorphic to a submodule of the other, then they must be isomorphic—indeed, any two such isomorphisms, from each of The research of the second and third authors was supported by grants from the Natural Sciences and Engineering Research Council of Canada. AMS 2000 Mathematics Subject Classification. Primary: 46L05, 46L35, 46M15;