Let
$G$
be a compact group. The aim of this note is to show that the only continuous *-homomorphism from
$L^{1}(G)$
to
$\ell ^{\infty }\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$
that transforms a convolution product into a pointwise product is, essentially, a Fourier transform. A similar result is also deduced for maps from
$L^{2}(G)$
to
$\ell ^{2}\text{-}\bigoplus _{[{\it\pi}]\in {\hat{G}}}{\mathcal{B}}_{2}({\mathcal{H}}_{{\it\pi}})$
.
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