On relative homological dimension of group algebras of locally compact groups

Let $G$ be a noncompact, locally compact group with an invariant mean, $L_1(G)$ its group algebra, and $I$ the ideal of $L_1(G)$ formed by those functions whose Haar integral is zero.
In this paper it is shown that the (relative) homological dimension of the Banach $L_1(G)$-module $L_1(G)/I$ is infinite. By the same token the (relative) global dimension of the Banach algebra $L_1(G)$ is also infinite. This result is then applied to the study of cohomology groups of a locally compact group with coefficients in Banach $G$-modules.