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Approximating convolution with $L^1$ function by sum of translation operators

Mathematics Asked by clhpeterson on December 2, 2020

Let $G$ be a locally compact topological group. Let $mu$ be left Haar measure on G. Suppose $f in L^1(G)$. Then convolution with $f$ defines an operator from left uniformly continuous and bounded functions ($UCB(G)$) to itself. I am wondering if it is always true that given $phi in UCB(G)$, and $varepsilon > 0$, there exists finitely many values $c_i in mathbb{C}$ and points $g_i in G$ such that $|sum c_i {}_{g_i} phi – f * phi|_infty < varepsilon$. Here
$${}_{g_i}phi (x) = phi(g_i x)$$
and
$$f * phi (x) := int_{G} f(y) {}_{y^{-1}}phi(x) d mu(y).$$

For context, in the book I’m reading (Property (T) by Bekka, De La Harpe, and Valette), in the section about amenability, they define amenability to mean the existence of a (finitely additive) invariant mean on $UCB(G)$, and then later claim that if $m$ is such a mean, then for any $f in L^1(G)$ with $f geq 0$ and $|f|_1 = 1$ and $phi in UCB(G)$, $m(f * phi) = m (phi)$. This result intuitively makes sense to me, but I am failing to see how to rigorously show this.

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