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

TL;DR

This paper characterizes the conditions under which Banach and $C^*$-algebras from continuous fields are projective or biprojective, linking algebraic properties to topological and spectral conditions.

Contribution

It provides necessary and sufficient conditions for projectivity of Banach and $C^*$-algebras constructed from continuous fields, including spectral characterizations.

Findings

Left projectivity of certain $C^*$-algebras is equivalent to the existence of a strictly positive element.

For commutative $C^*$-algebras, projectivity corresponds to the spectrum being a Lindelöf space.

The paper identifies classes of $C^*$-algebras with projective properties based on their structure and spectrum.

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.