Characterizations of $B(G)$ and $B(G)\cap AP(G)$ for locally compact groups

Abstract:

Given a locally compact (and possibly non-Abelian) group $G$, we denote by $B(G)$ the set of linear combinations of continuous positive-definite functions on $G$ and by $AP(G)$ the set of continuous almost periodic functions on $G$. In this paper the sets $B(G)$ and $B(G) \cap AP(G)$ are characterized in terms of convolutions with measures. Specifically, let $U$ consist of those measures $\mu \in M(G)$ for which $||\pi (\mu )|| \leq 1$, whenever $\pi$ is a continuous unitary representation of $G$. It is proved that a function $f \in {L^\infty }(G)$ belongs to (i.e. is equal locally almost everywhere to a function in) $B(G)$ if and only if the convolutions $\mu \ast f,\mu$ ranging over $U$, form a relatively weakly compact set in ${L^\infty }(G)$. The same holds if we confine our attention to either the finitely supported or the absolutely continuous measures in $U$. Moreover, it is shown that any of these three sets of convolutions is relatively norm compact if and only if $f$ belongs to $B(G) \cap AP(G)$.