Definition 30.4.7 ($L^{1}$ Group Algebra).label Let $G$ be a locally compact group, then $L^{1}(G; \complex)$ equipped with the convolution product is an involutive Banach subalgebra of $M_{R}(G; \complex)$, where for any $f \in L^{1}(G; \complex)$,
\[f^{*}(x) = \Delta(x^{-1}) \ol{f(x^{-1})}\]
The algebra $L^{1}(G; \complex)$ is the $L^{1}$ group algebra of $G$.
Proof. By Lemma 30.4.6, the convolution operation on $L^{1}(G; \complex)$ agrees with the convolution on $M_{R}(G; \complex)$. By Young’s Inequality, $L^{1}(G; \complex)$ is a Banach subalgebra of $M_{R}(G; \complex)$.
For any $f \in L^{1}(G; \complex)$, let $\mu = f dx$, then for each $A \in \cb_{G}$,
\[\mu^{*}(A) = \ol{\mu(A^{-1})}= \ol{\int_{A^{-1}}d\mu}= \int_{A^{-1}}\ol{f(x)}dx = \int_{A} \ol{f(x^{-1})}\Delta_{G}(x^{-1})dx\]
so $f^{*}dx = d\mu^{*} = \Delta(x^{-1}) \ol{f(x^{-1})}dx$, and $f^{*} = \Delta(x^{-1}) \ol{f(x^{-1})}$.$\square$
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