Projectivity of Banach and $C^*$-algebras of continuous fields

Details Projectivity of Banach and $C^*$-algebras of continuous fields Projectivity of Banach and $C^*$-algebras of continuous fields

2011-04-26

arxiv.org/abs/1104.4935v1

Mathematics

Functional Analysis

34 pages

Scientific paper

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.

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