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