Definition 44.2.5 (Space of Left Convolvers).label Let $G$ be a group, $LC(G) := \bracsn{L_f| f: G \to \complex\text{ is a left convolver}}\subset B(l^{2}(G; \complex))$, and $f, g: G \to \complex$ be left convolvers, then:
- (1)
$f \in l^{2}(G; \complex)$.
- (2)
$L_{f}^{*} = L_{f^*}$, where $f^{*}(x) = \ol{f(x^{-1})}$ for all $x \in G$.
- (3)
$L_{f * g}= L_{f}L_{g}$.
- (4)
$LC(G) \subset B(l^{2}(G; \complex))$ is a von Neumann algebra.
The set $LC(G)$ is the algebra of left convolvers on $G$.
Proof, [Proposition 1.3.5, AP]. (1): $f * \delta_{e} = f \in l^{2}(G; \complex)$.
(2): Let $x, y \in G$, then
(3): By (1), $f * g \in l^{2}(G; \complex)$ is defined. For any $h \in l^{2}(G; \complex)$, by associativity of convolution,
(4): Let $T \in \ol{LC(G)}^{\text{\small SOT}}$ and $\angles{f_\alpha}_{\alpha \in A}$ be left convolvers such that $T = \sotlim_{\alpha \in A}L_{f_\alpha}$. Let $f = T\delta_{e} \in l^{2}(G; \complex)$, then
For any $g \in l^{2}(G; \complex)$,
by Young’s Inequality. Therefore $Tg = f * g$, $T = L_{f}$, and $LC(G)$ is strong-operator closed. By von Neumann’s Bicommutant Theorem, $LC(G)$ is a von Neumann algebra.$\square$
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