# 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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