Let $G$ be a locally compact abelian Hausdorff group (with Haar measure $d\mu$). Call a function $h \in L^\infty(G)$ a function of positive type if $$ \int_G (f^* *f)h \, d\mu \geq 0 \ \ \ \forall f \in L^1(G) \ \ \ \ \ \ (\#) $$

where $f^*(x) := \overline{f(x^{-1})}$ denotes the involution.

Now let $g \in L^1(G) \cap L^2(G)$. I want to show that $h := g^**g$ is of positive type. Clearly $h \in L^\infty(G)$ by Cauchy Schwarz but I have trouble to show the property $(\#)$.


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