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

Author: Florin Radulescu
Journal: Proc. Amer. Math. Soc. 128 (2000), 2405-2411
MSC (1991): Primary 46L35; Secondary 46L37, 46L57, 81S99, 11F99
Published electronically: November 29, 1999
MathSciNet review: 1662202
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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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Additional Information

Florin Radulescu
Affiliation: Department of Mathematics, The University of Iowa, Iowa City, Iowa 52246

Received by editor(s): September 22, 1998
Published electronically: November 29, 1999
Additional Notes: The author’s research was supported in part by the grant DMS 9622911 from the National Science Foundation. The author is a member of the Institute of Mathematics, Romanian Academy, Bucharest.
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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