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If $G$ is a compact group, and we choose the normalized Haar measure $\mu$ on $G$, then $L^2(G)$ is a Banach algebra with convolution. Indeed, if $f$ and $h$ are in $L^2(G)$, then we use Hölder's inequality at the second step, together with the fact that the constant function 1 has $L^2$-norm equal to 1, to compute \begin{align*} |f\ast h(x)|\leq \int_G|f(y)h(x-y)|1\ d\mu(y)\leq \|f\|_2\|h\|_2. \end{align*} Thus $$\|f\ast h\|_2\leq \|f\ast h\|_{\infty}\leq \|f\|_2 \|h\|_2.$$ This Banach $\ast$-algebra is not only reflexive, but it is also a Hilbert space.