Maximal ideals in the group algebra of an extension, with applications

Abstract:

Let $A$ be a closed central subgroup of $E$ (all groups are locally compact and second countable). Let $G = E/A$. For each $a\in \hat A$, the dual of $A$, a multiplication is introduced with respect to which the Banach space ${L^1}(G)$ is a Banach algebra, denoted by ${L^1}(G,a(\sigma ))$. A one-to-one correspondence is established between the maximal closed (right, left, $2$-sided) ideals of the group algebra ${L^1}(E)$ and the totality of maximal closed (right, left, $2$-sided) ideals of ${L^1}(G,a(\sigma ))$, where $a$ varies over $\hat A$. Applications include a bound for the spectral norm of an element of ${L^1}(E)$ and the representation of a continuous positive definite function on $E$ as an integral (a ’Bochner’ theorem).