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