Definition 1

A bounded measure $\nu \in \mathscr{M}^1(G)$ is said to be central if one has $\varepsilon_g*\nu =\nu *\varepsilon_g$ for every $g\in G$.

If G is unimodular and $f\in \mathscr{C}(G)$ is $\mu$-integrable, the measure $f\cdot \mu$ is central if and only if $f$ is central.

Let $\varrho$ be a bounded continuous representation of G in a Banach space E (resp. a continuous representation of G in a separated quasi-complete locally convex space E). For every bounded central measure $\nu$ on G (resp. every central measure with compact support $\nu$ on G), the linear mapping $\varrho (\nu )$ is a G-morphism from $\varrho$ into $\varrho$. In fact, for every $g\in G$, one has