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

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.