Homological properties of the algebra of compact operators on a Banach space

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.