Definition 44.2.2 (Left Convolver).label Let $G$ be a locally compact group and $f: G \to \complex$ be Borel measurable, then $f$ is a left convolver if for each $g \in L^{2}(G; \complex)$, $\int_{G} |f(y)||g(y^{-1}x)|dy < \infty$ for almost every $x \in G$ and $f * g \in L^{2}(G; \complex)$. The mapping $L_{f}: L^{2}(G; \complex) \to L^{2}(G; \complex)$ defined by $g \mapsto f * g$ is the left convolution map by $f$.