Proposition 6

The mapping $m:g\mapsto \widetilde{m}_g$ is an isometric unital morphism of the involutive algebra $L^{\infty}(X, \mu)$ into the involutive algebra $\mathscr{L}(L^2(X, \mu))$.

In particular, for every $g\in L^{\infty}(X, \mu)$, the multiplication endomorphism $\widetilde{m}_g$ is normal; it is hermitian if and only if $g$ is locally $\mu$-almost everywhere real-valued.

By Prop. 5, the mapping $\widetilde{m}$ is an injective isometric morphism of unital Banach algebras of $L^{\infty}(X, \mu)$ into $\mathscr{L}(L^2(X, \mu))$. Let $g\in \mathscr{L}^{\infty}(X, \mu)$. For $f_1$ and $f_2\in \mathscr{L}^2(X, \mu)$, one has

whence it follows that $\widetilde{m}^*_g=\widetilde{m}_{\overline{g}}$, which proves that $m$ is an involutive morphism. The final assertions follow from this (cf. I, p. 106, Prop. 5).