Definition 44.2.1 (Group von Neumann Algebra).label Let $G$ be a locally compact group,
\[\lambda: G \to U(L^{2}(G; \complex)) \quad x \mapsto L_{x}\]
be the left regular representation of $G$, and
\[\lambda: L^{1}(G; \complex) \to B(L^{2}(G; \complex)) \quad \lambda(f)(g) = (f * g)\]
be its extension to $L^{1}(G; \complex)$, then
\[L(G) := \lambda(G)'' = \lambda(L^{1}(G;\complex))''\]
is the group von Neumann algebra of $G$. Similarly, if $\rho: G \to U(L^{2}(G; \complex))$ is the right regular representation of $G$, then $R(G) := \rho(G)'' = \rho(L^{1}(G;\complex))''$.
Proof. Firstly, for each $f \in L^{1}(G; \complex)$ and $g, h \in L^{2}(G; \complex)$, by the Fubini-Tonelli Theorem,
\begin{align*}\dpn{\lambda(f)g, h}{L^2(G; \complex)}&= \int_{G} f(x)\dpn{L_xg, h}{L^2(G; \complex)}dx = \iint_{G \times G}f(x) g(x^{-1}y) \ol{h}(y) dy dx \\&= \iint_{G \times G}f(x) g(x^{-1}y)dx \ol{h}(y) dy \\&= \int_{G} (f * g)(y) \ol{h}(y)dy = \dpn{(f * g), h}{L^2(G; \complex)}\end{align*}
By Theorem 32.2.2, $\ol{\lambda(G)}^{\text{\small SOT}}= \ol{\lambda(L^1(G;\complex))}^{\text{\small SOT}}$.$\square$
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