## Relations between the homologies of C*-algebras and their commutative C*-subalgebras

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### Abstract

The paper concerns the identification of projective closed ideals of C*-algebras. We prove that, if a C*-algebra has the property that every closed left ideal is projective, then the same is true for all its commutative C*-subalgebras. Further, we say a Banach algebra A is hereditarily projective if every closed left ideal of A is projective. As a corollary of the stated result we show that no infinite-dimensional AW*-algebra is hereditarily projective. We also prove that, for a commutative C*-algebra A contained in B(H), where H is a separable Hilbert space, the following conditions are equivalent: (i) A is separable; and (ii) the C*-tensor product A circle times(min) A is hereditarily projective. Howerever, there is a non-separable, hereditarily projective, commutative C*-algebra A contained in B(H), where H is a separable Hilbert space.

### Publication metadata

**Author(s): **Lykova ZA

**Publication type: **Article

**Publication status:** Published

**Journal: **Mathematical Proceedings of the Cambridge Philosophical Society

**Year: **2002

**Volume: **132

**Pages: **155-168

**ISSN (print): **0305-0041

**ISSN (electronic): **1469-8064

**Publisher: **Cambridge University Press

**URL: **http://dx.doi.org/10.1017/S0305004101005497

**DOI: **10.1017/S0305004101005497

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