Regarding cyclic vector of $L^{\infty}$ in $L^{2}$

Question

What are cyclic vectors for $L^{\infty}([0,1],\mu)\otimes L^{\infty}([0,1],\mu)$ in $L^{2}([0,1],\mu)\otimes L^{2}([0,1],\mu)$ other than $1\otimes 1$? I want nonconstant cyclic vector in $f\otimes g$ form?

Answer 1

Take any $f,g\in L^\infty$ such that there exist $r,s>0$ with $$r\leq|f|\leq s,\ \ \ r\leq|g|\leq s.$$ Then $f,g$ are each cyclic for $L^\infty[0,1]$ and so $f\otimes g$ is cyclic for $L^\infty[0,1]\otimes L^\infty[0,1]$. It is trivial to verify this, since $1/f\otimes 1/g\in L^\infty[0,1]\otimes L^\infty[0,1]$, so $$ (L^\infty[0,1]\otimes L^\infty[0,1])\,(f\otimes g)=L^\infty[0,1]\otimes L^\infty[0,1]. $$