Injective and projective Hilbert C*-modules, and C*-algebras of compact operators
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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.
28 pages, Proposition 3.4 corrected after receiving remarks from Hanfeng Li, thanks for pointing out the errors. Some minor corrections in formulations. 15 references added
28 pages, Proposition 3.4 corrected after receiving remarks from Hanfeng Li, thanks for pointing out the errors. Some minor corrections in formulations. 15 references added