Injective and projective Hilbert C*-modules, and C*-algebras of compact operators

TL;DR

This paper characterizes C*-subalgebras of compact operators via properties of Hilbert C*-modules, establishing conditions for injectivity and projectivity in various module categories, and extends known results in the field.

Contribution

It provides new characterizations of C*-subalgebras of compact operators using Hilbert C*-modules and analyzes their injectivity and projectivity properties.

Findings

Hilbert C*-modules over compact operator C*-algebras are both injective and projective

Characterization of C*-subalgebras of compact operators via module properties

Identification of classes of injective and projective modules over general C*-algebras

Abstract

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.