Injective and Projective Hilbert C*-modules, and C*-algebras of Compact Operators

We consider projectivity and injectivity of Hilbert C*-modules in the categories of Hilbert C*-(bi-)modules over a fixed C*-algebra of coefficients (and another fixed C*-algebra represented as bounded module operators) and bounded (bi-)module morphisms, either necessarily adjointable or arbitrary ones. As a consequence of these investigations, we obtain a set of equivalent conditions characterizing C*-subalgebras of C*-algebras of compact operators on Hilbert spaces in terms of general properties of Hilbert C*-modules over them. Our results complement results recently obtained by B. Magajna, J. Schweizer and M. Kusuda. In particular, all Hilbert C*-(bi-)modules over C*-algebras of compact operators on Hilbert spaces are both injective and projective in the categories we consider. For more general C*-algebras we obtain classes of injective and projective Hilbert C*-(bi-)modules. The goal of this paper is to determine the injective and projective Hilbert C*-modules over a fixed C*-algebra, when one allows the maps between C*-modules to be bounded. Most prior work on injectivity has focused on the case of contractive maps. To better understand our motivations and the distinction between this work and the work of others, we first review the concept of injectivity and some history of the subject. To give a definition of the term, injective, that is useful for our purposes, we need a category, consisting of objects that are sets and morphisms between them that are functions, and for each object, N , certain subsets, M ⊆ N that are also objects in the category, which we call the subobjects of N . Then an object I in this category is called injective, provided that for every object N , every subobject, M ⊆ N and every morphism, φ : M → I, there is a morphism ψ : N → I, that extends φ. Note that if we keep the objects and morphisms the same, 1991 Mathematics Subject Classification. Primary 46L08 ; Secondary 46H25.