Let $G$ be a locally compact topological amenable group and $\{\pi, H\}$ a representation of $M(G)$, then

$$\text{Inf} \ \{\Vert \pi(\mu)x\Vert : \mu \in M(G) \} = \text{dis}(x, K_{H,\pi}) \; \text{for all} \; x \in H$$

where $K_{H,\pi}$ is the linear span of $\{y-\pi(\mu)y: y \in H, \mu \in M_0(G)\}$ and $\text{dis}(x, K_{H,\pi})$ is the distance of $x$ from $K_{H,\pi}$. If $G$ is a nontrivial compact group then the regular representation is reducible, also $P: L^2(G) \to \overline{K_{\pi,H}}$ the orthogonal projection of $L^2(G)$ onto $\overline{K_{H,\pi}}$ commutes with convolution, that is $P(\mu * f) = \mu*Pf$ for all $\mu \in M(G)$ and all $f$ in $L^2(G)$.