Exercise 22, § 2

Let G be a locally compact commutative topological group and $\mu$ a Haar measure on G.

Let $u\in \mathscr{L}(L^2(G, \mu))$ be such that $u$ commutes with the multiplication operator $m_{\chi}$ for every character $\chi \in \widehat{G}$. Prove that there exists $g\in L^{\infty}(G, \mu)$ such that $u=m_g$.