Proving difficulty in MASA

Question

Prove that $ L^\infty{(\mathbb{R},\mu)}$ is a masa in $B(L^2(\mathbb{R},\mu))$, $\mu$ is sigma finite measure in particular Lebesgue, and what is the cyclic vector of it?

Answer 1

Any function that is essentially nonzero is both cyclic and separating. In particular this proves that $L^\infty(\mathbb R,\mu)$ is a masa.

For instance, let $$ f(t)=\sum_{n\in\mathbb Z} \tfrac1{|n|}\,1_{[n,n+1]}. $$ Then, for any $g\in L^2$ bounded with compact support contained in $[-m,m]$, let $$ h=\sum_{n=-m}^m|n|\,g\,1_{[n,n+1]}\in L^\infty; $$ then $hf=g$. As bounded functions of compact support are dense in $L^2$, $f$ is cyclic. And if $h\in L^\infty$ and $hf=0$, then $h=0$ a.e., so $f$ is separating.