Proposition 7

The image of the morphism $\widetilde{m}$ from $L^{\infty}(X, \mu)$ into $\mathscr{L}(L^2(X, \mu))$ is a maximal commutative subalgebra of the algebra $\mathscr{L}(L^2(X, \mu))$.

It is enough to prove that if $u\in \mathscr{L}(L^2(X, \mu))$ is an endomorphism which commutes with $\widetilde{m}_g$ for every function $g\in \mathscr{L}^{\infty}(X, \mu)$, then $u$ belongs to the image of $\widetilde{m}$. Let $u$ be such an endomorphism.

We denote by $\widetilde{f}$ the class in $L^2(X, \mu)$ of a function $f\in \mathscr{L}^2(X, \mu)$. Let $v$ denote the linear mapping from $\mathscr{L}^2(X, \mu)$ into $L^2(X, \mu)$ defined by $f\mapsto u(\widetilde{f})$. One has $v\circ m_g=\widetilde{m}_g\circ v$ for every $g\in \mathscr{L}^{\infty}(X, \mu)$.

For every μ-measurable subset Y of X, let $\varphi_Y$ denote its characteristic function; the endomorphism $\widetilde{m}_{\varphi_Y}$ is an orthogonal projector of $L^2(X, \mu)$.

Suppose first that X is compact, so that the class of the constant function 1 belongs to $L^2(X, \mu)$. Let $g$ be a function in $\mathscr{L}^2(X, \mu)$ whose class in $L^2(X, \mu)$ is $v(1)$.

Let $c$ be a positive real number and Y the set of $x\in X$ such that $|g(x)|\geqslant c$; this is a μ-measurable set. One has in $L^2(X, \mu)$ the equalities $\varphi_{\widetilde{Y}}g=\widetilde{m}_{\varphi_Y}(v(1)) =v(m_{\varphi_Y}(1)) =v(\varphi_Y)$.

Since moreover $|c\varphi_Y|\leqslant |g\varphi_Y|$, one obtains

This inequality implies that $\mu(Y) = 0$ if $c >\|u\|$, so that the function $g$ is bounded μ-almost everywhere by $\|u\|$. Finally, for every function $f\in \mathscr{C}$(X), one has

whence the equality $u=\widetilde{m}_g$, and in particular $N_{\infty}(g) =\|u\|$.

Let us now consider the general case. There exists a locally countable family $(X_i)_{i\in I}$ of pairwise disjoint compact subsets of X such that Z = X $-\bigcup_{i\in I}X_i$ is locally $\mu$-negligible (INT, IV, p .190, § 5, n$^o9$, prop. 14). Let $\mu_i$ denote the measure induced by $\mu$ on $X_i$ (INT, IV, p. 186, § 5, n$^o7$, def. 4).

By prop. 1 of IV, p. 179, for every $i\in I$, the image $E_i$ of $\widetilde{m}_{\varphi_{Xi}}$ is identified with $L^2(X_i, \mu_i)$ by $f\mapsto f|X_i$. For every function $g\in \mathscr{L}^{\infty}(X, \mu)$, the multiplication endomorphism $\widetilde{m}_g$ commutes with $\widetilde{m}_{\varphi_{Xi}}$, hence leaves $E_i$ stable, and the endomorphism of $E_i$ deduced from $\widetilde{m}_g$ by passing to subspaces coincides with the multiplication endomorphism by $g|X_i$ on $L^2(X_i, \mu_i)$.

Since $u$ commutes with $m_{\varphi_{Xi}}$, it likewise leaves the subspace $E_i$ stable. Let $u_i$ denote the endomorphism of $L^2(X_i, \mu_i)$ deduced from $u$ by passing to subspaces.

The map from $\mathscr{L}^{\infty}(X, \mu)$ into $\mathscr{L}^{\infty}(X_i, \mu_i)$ defined by $g\mapsto g|X_i$ being surjective, the hypothesis implies that $u_i$ commutes with $\widetilde{m}_g$ for every function $g\in \mathscr{L}^{\infty}(X_i, \mu_i)$. By the first part of the proof, there exists a function $g_i\in \mathscr{L}^{\infty}(X_i, \mu_i)$ such that $u_i=\widetilde{m}_{g_i}$ and $N_{\infty}(g_i)\leqslant \|u_i\|\leqslant \|u\|$.

The function $g$ on X which coincides with $g_i$ on $X_i$ and which is zero on Z is bounded and $\mu$-measurable (INT, IV, p. 193, § 5, n$^o10$, prop. 16), hence $g\in \mathscr{L}^{\infty}(X, \mu)$. For every $i\in I$, the restriction of $u$ to $E_i$ coincides with that of $\widetilde{m}_g$; therefore one has $u=m_g$ by prop. 1, c) of IV, p. 179.