= Integrated unitary representation
{title2=$\pi(f)=\int_G f(x)U_x\,dx$}
For a strongly continuous <unitary representation>, integration against $f\in L^1(G)$ defines an operator satisfying $\|\pi(f)\|\le\|f\|_1$. <Fubini's theorem> gives $\pi(f*g)=\pi(f)\pi(g)$, and the group-algebra involution gives $\pi(f^*)=\pi(f)^*$. On an Abelian group, the operator-norm closure is a commutative <C-star algebra>. Its <algebra characters> produce continuous unitary <group characters>, allowing the <Commutative Gelfand--Naimark theorem> and <Riesz-Markov-Kakutani representation theorem> to turn a positive vector functional into a finite spectral measure.
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