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).

Rădulescu, F. (2000). Finite generation properties for fuchsian group von Neumann algebras tensor. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 128(08), 2405-2412 [10.1090/S0002-9939-99-05308-3].

### Finite generation properties for fuchsian group von Neumann algebras tensor

#### Abstract

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).
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http://www.ams.org/journals/proc/2000-128-08/S0002-9939-99-05308-3/S0002-9939-99-05308-3.pdf
Rădulescu, F. (2000). Finite generation properties for fuchsian group von Neumann algebras tensor. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 128(08), 2405-2412 [10.1090/S0002-9939-99-05308-3].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/100656