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/Part 4: Measure Theory and Integration/Chapter 30: Integration on Locally Compact Groups/Section 30.4: The $L^{1}$ Group Algebra

Definition 30.4.3 (Measure Algebra).label Let $G$ be a locally compact group. For each $\mu \in M_{R}(G; \complex)$, let

\[\mu^{*}: \cb_{G} \to \complex \quad \mu^{*}(A) = \ol{\mu(A^{-1})}\]

then $M_{R}(G; \complex)$ equipped with the convolution product and the above involution is an involutive unital Banach algebra with identity $\delta_{1}$. The algebra $M_{R}(G; \complex)$ is the measure algebra of $G$.

Proof. By (2) of Unknown (lemma:convolution-measures-property), $M_{R}(G; \complex)$ with the convolution product is a Banach algebra. For any $\mu \in M_{R}(G; \complex)$ and $\phi \in C_{c}(G; \complex)$,

\begin{align*}\int_{G} \phi d(\mu * \delta_{1})&= \iint_{G \times G}\phi(xy)\mu(dx)\delta_{1}(dy) \\&= \iint_{G \times G}\phi(xy)\delta_{1}(dy)\mu(dx) = \int_{G} \phi(x)\mu(dx) \\ \int_{G} \phi d(\delta_{1} * \mu)&= \iint_{G \times G}\phi(xy)\delta_{1}(dx)\mu(dy) = \int_{G} \phi(y)\mu(dy)\end{align*}

so $\mu * \delta_{1} = \mu = \delta_{1} * \mu$, and $\delta_{1}$ is the identity.

It remains to show that $\mu \mapsto \mu^{*}$ is an involution. Let $\mu, \nu \in M_{R}(G; \complex)$, then for any $\phi \in C_{c}(G; \complex)$,

\begin{align*}\int_{G} \phi d(\mu * \nu)^{*}&= \int_{G} \phi(x^{-1})d(\ol{\mu * \nu}) = \iint_{G \times G}\phi((xy)^{-1})\ol{\mu}(dx)\ol{\nu}(dy) \\&= \iint_{G \times G}\phi(y^{-1}x^{-1}) \ol{\mu}(dx)\ol{\nu}(dy) = \iint_{G \times G}\phi(yx)\mu^{*}(dx)\nu^{*}(dy) \\&= \iint_{G \times G}\phi(yx)\nu^{*}(dy)\mu^{*}(dx) = \int_{G}\phi d(\nu^{*} * \mu^{*})\end{align*}

Therefore $(\mu * \nu)^{*} = \nu^{*} * \mu^{*}$.$\square$

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