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

Michael Frank; Vern I. Paulsen

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

Operator Algebras 2008-02-18 v2 K-Theory and Homology

Authors: Michael Frank , Vern I. Paulsen

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.

Keywords

injective hilbert c*-modules projective hilbert c*-modules c*-algebras of compact operators adjointable morphisms bounded module operators equivalent conditions

Cite

@article{arxiv.math/0611349,
  title  = {Injective and projective Hilbert C*-modules, and C*-algebras of compact operators},
  author = {Michael Frank and Vern I. Paulsen},
  journal= {arXiv preprint arXiv:math/0611349},
  year   = {2008}
}

Comments

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