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Projectivity of Banach and C*-algebras of continuous fields

Lookup NU author(s): Dr David Cushing, Dr Zinaida LykovaORCiD

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Abstract

We give necessary and sufficient conditions for the left projectivity and biprojectivity of Banach algebras defined by locally trivial continuous fields of Banach algebras. We identify projective $C^*$-algebras $\A$ defined by locally trivial continuous fields $\mathcal{U} = \{\Omega,(A_t)_{t \in \Omega},\Theta\}$ such that each $C^*$-algebra $ A_{t}$ has a strictly positive element. For a commutative $C^*$-algebra $\D$ contained in ${\cal B}(H)$, where $H$ is a separable Hilbert space, we show that the condition of left projectivity of $\D$ is equivalent to the existence of a strictly positive element in $\D$ and so to the spectrum of $\D$ being a Lindel$\ddot{\rm o}$f space.

Publication metadata

Author(s): Cushing D, Lykova ZA

Publication type: Article

Publication status: Published

Journal: Quarterly Journal of Mathematics

Year: 2013

Volume: 64

Issue: 2

Pages: 341-371

Print publication date: 19/03/2012

Date deposited: 28/04/2011

ISSN (print): 0033-5606

ISSN (electronic): 1464-3847

Publisher: Oxford University Press

URL: http://dx.doi.org/10.1093/qmath/has005

DOI: 10.1093/qmath/has005

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