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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

Proposition 30.4.2.label Let $G$ be a locally compact group, then:

  1. (1)

    For any $\mu, \nu, \sigma \in M_{R}(G; \complex)$, $(\mu * \nu) * \sigma = \mu * (\nu * \sigma)$.

  2. (2)

    For any $\mu, \nu \in M_{R}(G; \complex)$, $\norm{\mu * \nu}_{\text{var}}\le \norm{\mu}_{\text{var}}\norm{\nu}_{\text{var}}$.

  3. (3)

    Convolution on $M_{R}(G; \complex)$ is commutative if and only if $G$ is commutative.

Proof. (1): Let $\phi \in C_{c}(G; \complex)$, then

\begin{align*}\int_{G} \phi d(\mu * (\nu * \sigma))&= \iint_{G \times G}\phi(xy)\mu(dx)(\nu * \sigma)(dy) \\&= \iiint_{G \times G \times G}\phi(xyz) \mu(dx) \nu(dy)\sigma(dz) \\&= \iint_{G \times G}\phi(yz) (\mu * \nu)(dy) \sigma(dz) = \int_{G} \phi d((\mu * \nu) * \sigma)\end{align*}

(2): Let $\phi \in C_{c}(G; \complex)$, then

\begin{align*}\abs{\int_G \phi d(\mu * \nu)}&= \abs{\iint_{G \times G}\phi(xy)\mu(dx)\nu(dy)}\\&\le \abs{\int_{G}\norm{\phi}_u\norm{\mu}_{\text{var}}d|\nu|}\le \norm{\phi}_{u}\norm{\mu}_{\text{var}}\norm{\nu}_{\text{var}}\end{align*}

(3): For any $g, h \in G$ and $\phi \in C_{c}(G; \complex)$,

\[\int_{G} \phi d(\delta_{g} * \delta_{h}) = \iint_{G \times G}\phi(xy) \delta_{g}(dx)\delta_{h}(dy) = \phi(gh)\]

so $\delta_{g} * \delta_{h} = \delta_{gh}$, and $\delta_{g} * \delta_{h} = \delta_{h} * \delta_{g}$ if and only if $gh = hg$. Therefore the convolution in $M_{R}(G; \complex)$ is commutative if and only if $G$ is commutative.$\square$

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