process

Categorical aspects of bivariant K-theory 15in a suitable sense. Already the simplest possible case X D G shows that we cannotexpect an isomorphism here becauseG Ë C 0 .G/ Š G Ë r C 0 .G/ Š K.L 2 G/:The right notion of equivalence is a C -version of Morita equivalence due to Marc A.Rieffel ([46], [47], [48]); therefore, we call it Morita–Rieffel equivalence.The definition of Morita–Rieffel equivalence involves Hilbert modules over C -algebrasand the C -algebras of compact operators on them; these notions are crucial forKasparov theory as well. We refer to [34] for the definition and a discussion of theirbasic properties.Definition 29. Two G-C -algebras A and B are called Morita–Rieffel equivalent ifthere are a full G-equivariant Hilbert B-module E and a G-equivariant -isomorphismK.E/ Š A.It is possible (and desirable) to express this definition more symmetrically: E is anA; B-bimodule with two inner products taking values in A and B, satisfying variousconditions [46]. Morita–Rieffel equivalent G-C -algebras have equivalent categoriesof G-equivariant Hilbert modules via E ˝B . The converse is unclear.Example 30. The following is a more intricate example of a Morita–Rieffel equivalence.Let and P be two subgroups of a locally compact group G. Then acts onG=P by left translation and P acts on nG by right translation. The correspondingorbit space is the double coset space nG=P . Both ËC 0 .G=P / and P ËC 0 .nG/arenon-commutative models for this double coset space. They are indeed Morita–Rieffelequivalent; the bimodule that implements the equivalence is a suitable completion ofC c .G/, the space of continuous functions with compact support on G.These examples suggest that Morita–Rieffel equivalent C -algebras describe thesame non-commutative space. Therefore, we expect that reasonable functors on C algshould not distinguish between Morita–Rieffel equivalent C -algebras. (We willslightly weaken this statement below.)Definition 31. Two G-C -algebras A and B are called stably isomorphic if there is aG-equivariant -isomorphism A ˝ K.H G / Š B ˝ K.H G /, where H G WD L 2 .G N/is the direct sum of countably many copies of the regular representation of G; we let Gact on K.H G / by conjugation, of course.The following technical condition is often needed in connection with Morita–Rieffelequivalence.Definition 32. AC -algebra is called -unital if it has a countable approximate identityor, equivalently, contains a strictly positive element.Example 33. All separable C -algebras and all unital C -algebras are -unital; thealgebra K.H/ is -unital if and only if H is separable.