Finite Generation Type Properties for Fuchsian Group Von Neumann Algebras

In this paper we prove the existence of finite systems G of generators, with special properties, for the von Neumann algebras L(Γ)⊗B(H), obtained by tensoring the group von Neumann algebra of a fuchsian group Γ with the bounded linear operators on a infinite dimensional, separable, Hilbert space H . The results show that the linear span of ordered monomials that are products of powers of adjoints of elements in G with powers of elements in G, is weakly dense in the algebra. In addition the systems of generators are a commuting family, having non-trivial, joint invariant subspaces affiliated to the von Neumann algebra. If Γ = PSL(2,Z), we can find such a generating set consisting of a single element. The construction in this paper may also be used to provide examples for Toeplitz operator with unbounded symbol with unexpected behaviour and nonclosability for multiplication operators, with bounded symbol, in Sobolev type Hilbert spaces associated to differential operators. We recall some of the notations that were used in [Ra3]. We let Ht be the Hilbert space of square summable, analytic functions on H, with respect to the measure dνt = (Im z) dzdz. In [Pu], [Sa] it was proven that there exist a one parameter family of irreducible, projective unitary representation of PSL(2,R) on Ht that extends the analytic discrete series of representations for PSL(2,R). Moreover the methods in [VFR] along with the trace formula in [Pu], were used in ([Ra3]) to show that for any fuchsian group Γ, the von Neumann algebra {πt(Γ} ′′ is a type II factor acting on Ht. The Murray von Neumann dimension for this algebra acting on Ht is proportional to t and the covolume of Γ. In particular the commutant algebras At = {πt(Γ}” are isomorphic to the twisted group von Neumann algebra L(Γ, σ)(t−1)/covolume Γ, with σ the cocycle corresponding to the projective representation πt. As shown in [GHJ], automorphic forms f of weight k for Γ correspond to bounded intertwining operators S f from Ht into Ht+k, by multiplication with f on Ht. The intertwining property means that πt+k(γ)S t f = S t fπt(γ), γ ∈ Γ.