= Convolution proof of uniform Peter-Weyl approximation
A continuous symmetric approximate identity on a <compact group> gives a compact self-adjoint <convolution> operator $T$ on $L^2(G)$ which maps into $C(G)$ and commutes with <left translations>. Its nonzero <eigenspaces> are finite-dimensional invariant spaces of <continuous functions>, hence consist of <matrix coefficients> of finite-dimensional <unitary representations>. If $S_N$ projects onto increasing finite sums of these <eigenspaces>, $S_NTf\to Tf$ in $L^2$, and the bound $\|Th\|_\infty\leq\|k\|_2\|h\|_2$ gives $TS_NTf\to T^2f$ uniformly. Choosing the approximate identity narrow makes $T^2f$ uniformly close to $f$. This supplies uniform approximation without confusing an $L^2$ limit with a uniform limit.
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