Let G be a locally compact group, and let be a lattice in G. Let (π, H π ) be an irreducible, square integrable unitary representation of G. Then the restriction of π to extends to V N ( ), the von Neumann algebra generated by the regular representation of . We determine the center valued (or extended) von Neumann dimension of H π as a V N ( )-module. This result is extended to representations which are square integrable modulo the center of G. An explicit formula is given when G is a semisimple algebraic group and when G is a nilpotent Lie group.As an application, we study the question of the existence of frames for the discretized windowed Fourier transform.

For n ≥ 3, let Γ = SL n (Z). We prove the following superridigity result for Γ in the context of operator algebras. Let L(Γ) be the von Neumann algebra generated by the left regular representation of Γ. Let M be a finite factor and let U (M ) be its unitary group. Let π : Γ → U (M ) be a group homomorphism such that π(Γ) ′′ = M. Then either (i) M is finite dimensional, or (ii) there exists a subgroup of finite index Λ of Γ such that π| Λ extends to a homomorphism U (L(Λ)) → U (M ).This answers, in the special case of SL n (Z), a question of A. Connes discussed in [Jone00, Page 86]. The result is deduced from a complete description of the tracial states on the full C * -algebra of Γ.As another application, we show that the full C * -algebra of Γ has no faithful tracial state, thus answering a question of E. Kirchberg.

A C *-algebra A is said to have the FS-property if the set of all self-adjoint elements in A has a dense subset of elements with finite spectrum. We shall show that this property is not stable under taking the minimal C *-tensor products even in case of separable nuclear C *-algebras. §1. Introduction A C *-algebra A is said to have the FS-property if the set of all self-adjoint elements in A (= A sa) has a dense subset of elements with finite spectrum. In [3], Brown and Pedersen provided the non-commutative real rank for A (= RR(A)) and showed that the FS-property is equivalent to real rank zero, i.e. A sa has a dense subset of invertible elements. RR(A) is the least integer n such that {(a 0 , a 1 , · · · , a n) ∈ A n+1 sa : n k=0 Aa k = A} is dense in A n+1 sa which is the analogue of the dimension of a topological space X(= dim X). If A is non-unital, its real rank is defined by RR(˜ A), where˜Awhere˜ where˜A is the C *-algebra obtained from A by adjoining a unit. From this definition it is obvious that dim X = RR(C(X)) for a compact Hausdorff space X. In the previous note [9], we showed that there are unital C *-algebras A and C with RR(A) = RR(C) = 0 such that RR(A ⊗ C) = 0. To specify let B be one of the Bunce-Deddens algebras and B(H) the C *-algebra of bounded operators on a countably infinite dimensional Hilbert space H. Then by Blackadar and Kumjian [1], RR(B) = 0 and it is well-known that RR(B(H)) = 0. In [9], we showed that RR(B ⊗ B(H)) = 0. This means that the FS-property for C *-algebras is not stable under taking the minimal C *-tensor products. But B(H) is not separable. In this note, we shall give two examples: Example 1. There are separable unital C *-algebras A and C with RR(A) = RR(C) = 0 such that A and C are nuclear and that RR(A ⊗ C) = 0. Example 2. There are separable unital C *-algebras A and C with RR(A) = RR(C) = 0 such that A is not nuclear but exact, C is nuclear, and that RR(A⊗C) = 0.

A group is irreducibly represented if it has a faithful irreducible unitary representation. For countable groups, a criterion for irreducible representability is given, which generalises a result obtained for finite groups by W. Gaschütz in 1954. In particular, torsionfree groups and infinite conjugacy class groups are irreducibly represented.We indicate some consequences of this for operator algebras. In particular, we charaterise up to isomorphism the countable subgroups ∆ of the unitary group of a separable infinite dimensional Hilbert space H of which the bicommutants ∆ ′′ (in the sense of the theory of von Neumann algebras) coincide with the algebra of all bounded linear operators on H.

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