Finite generation properties for fuchsian group von Neumann algebras tensor $B(H)$

We prove that the algebra $\mathcal{A}=\mathcal{L}(F_{N})\otimes B(H)$, $F_{N}$ a free group with finitely many generators, contains a subnormal operator $J$ such that the linear span of the set $\{(J^{*})^{n}J^{m}\vert n,m=0,1,2,...\}$ is weakly dense in $\mathcal{A}$. This is the analogue for the $II_{\infty }$ factor $\mathcal{L}(F_{N})\otimes B(H)$, $N$ finite, of a well known fact about the unilateral shift $S$ on a Hilbert space $K$: the linear span of all the monomials $(S^{*})^{n} S^{m}$ is weakly dense in $B(K)$.

We also show that for a suitable space $H^{2}$ of square summable analytic functions, if $P$ is the projection from the Hilbert space $L^{2}$ of all square summable functions onto $H^{2}$ and $M_{\overline{j}}$ is the unbounded operator of multiplication by $\overline{j}$ on $L^{2}$, then the (unbounded) operator $PM_{\overline{j}}(I-P)$ is nonzero (with nonzero domain).