### Projectivity of Banach and C*-algebras of continuous fields (2013)

Author(s): Cushing D, Lykova ZA

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.

• Date: 19-03-2012
• Journal: Quarterly Journal of Mathematics
• Volume: 64
• Issue: 2
• Pages: 341-371
• Publisher: Oxford University Press
• Publication type: Article
• Bibliographic status: Published

Keywords: $C^*$-algebras; projectivity; continuous fields; continuous Hochschild cohomology; Suslin condition.

 Dr Zinaida Lykova Reader in Pure Mathematics Email: zinaida.lykova@ncl.ac.uk Telephone: +44 (0) 191 208 6479