Inclusions of C*-algebras

We introduce three notions of inclusions of C*-algebras: with the ideal property, with the weak ideal property, and with topological dimension zero. We characterize these notions and we show that for an inclusion of C*-algebras, the ideal property the weak ideal property double right arrow topological dimension zero. We prove that any two of these three notions do not coincide in general, but they are all equivalent in many interesting cases. We show some permanence properties for these notions, and we prove that they behave well with respect to tensor products and crossed products by discrete (finite) groups, in many interesting cases. For example, we prove that if A subset of B is an inclusion of C*-algebras which has topological dimension zero and alpha: G -> Aut(B) is a strongly pointwise outer action of a finite group G on B and if A is alpha-invariant, then the inclusion of crossed products C & lowast;(G, A, alpha) subset of C & lowast;(G, B, alpha) has topological dimension zero. We show that for an inclusion of C*-algebras, the real rank zero (in the sense of Gabe and Neagu [5]) the ideal property, and that these two notions do not coincide in general.