Let $C{V}_p\left(F\right)$ be the left convolution operators on $L^p\left(G\right)$ with support included in F and $M_p\left(F\right)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $C{V}_p\left(F\right)$, $C{V}_p\left(F\right)/M_p\left(F\right)$ and $C{V}_p\left(F\right)/W$ are as big as they can be, namely have $l^{\infty }$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_p\left(F\right)$. Other subspaces of $C{V}_p\left(F\right)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

Let $\tau$ be a continuous unitary representation of the locally compact group $G$ on the Hilbert space $H_{\tau }$. Let $C_{\tau }^{*}[V{N}_{\tau }]$ be the $C^{*}[W^{*}]$ algebra generated by$$(L^1\left(G\right))\text{and}{M}_{\tau }(C_{\tau }^{*})=\left\{\phi \in V{N}_{\tau };\phi C_{\tau }^{*}+C_{\tau }^{*}\phi \subset C_{\tau }^{*}\right\}.$$The main result obtained in this paper is Theorem 1:If $G$ is $\sigma$-compact and $M_{\tau }(C_{\tau }^{*})=V{N}_{\tau }$ then supp $\tau$ is discrete and each $\pi$ in supp $\tau$ in CCR.We apply this theorem to the quasiregular representation $\tau ={\pi }_H$ and obtain among other results that $M_{{\pi }_H}(C_{{\pi }_H}^{*})=V{N}_{{\pi }_H}$ implies in many cases that $G/H$ is a compact coset space.

We compute the $K$-theory of $C^{*}$-algebras generated by the left regular representation of left Ore semigroups satisfying certain regularity conditions. Our result describes the $K$-theory of these semigroup $C^{*}$-algebras in terms of the $K$-theory for the reduced group $C^{*}$-algebras of certain groups which are typically easier to handle. Then we apply our result to specific semigroups from algebraic number theory.