Homological Properties of the Algebra of Compact Operators on a Banach Space

TL;DR

This paper investigates the homological properties of the algebra of compact operators on a Banach space, linking these properties to the approximation property and extending known classes of spaces.

Contribution

It characterizes when the algebra of compact operators is right flat or homologically unital, especially for spaces lacking the bounded compact approximation property.

Findings

Homological properties are closely related to the approximation property.

Extension of classes of spaces where these properties hold.

Connections between factorization in the algebra and homological conditions.

Abstract

The conditions on a Banach space, $E$, under which the algebra, $\mathcal{K}(E)$, of compact operators on $E$ is right flat or homologically unital are investigated. These homological properties are related to factorization in the algebra and it is shown that, for $\mathcal{K}(E)$, they are closely associated with the approximation property for $E$. The class of spaces, $E$, such that $\mathcal{K}(E)$ is known to be right flat and homologically unital is extended to include spaces which do not have the bounded compact approximation property.