124 research outputs found
Quasitraces on exact C*-algebras are traces
It is shown that all 2-quasitraces on a unital exact C*-algebra are traces.
As consequences one gets: (1) Every stably finite exact unital C*-algebra has a
tracial state, and (2) if an AW*-factor of type II_1 is generated (as an
AW*-algebra) by an exact C*-subalgebra, then it is a von Neumann II_1-factor.
This is a partial solution to a well known problem of Kaplansky. The present
result was used by Blackadar, Kumjian and R{\o}rdam to prove that RR(A)=0 for
every simple non-commutative torus of any dimension
Invariant subspaces of the quasinilpotent DT-operator
We previously introduced the class of DT--operators, which are modeled by
certain upper triangular random matrices, and showed that if the spectrum of a
DT-operator is not reduced to a single point, then it has a nontrivial, closed,
hyperinvariant subspace. In this paper, we prove that also every DT-operator
whose spectrum is concentrated on a single point has a nontrivial, closed,
hyperinvariant subspace. In fact, each such operator has a one-parameter family
of them. It follows that every DT-operator generates the von Neumann algebra
L(F_2) of the free group on two generators
Invariant subspaces of Voiculescu's circular operator
We show that Voiculescu's circular operator and, more generally, each
circular free Poisson operator has a continuous family of invariant subspaces
relative to the von Neumann algebra it generates. The proof relies on upper
triangular random matrix models and consequent realizations of these operators
as upper triangular matrices of free random variables.Comment: 45 page
The weak Haagerup property II: Examples
The weak Haagerup property for locally compact groups and the weak Haagerup
constant was recently introduced by the second author. The weak Haagerup
property is weaker than both weak amenability introduced by Cowling and the
first author and the Haagerup property introduced by Connes and Choda.
In this paper it is shown that a connected simple Lie group G has the weak
Haagerup property if and only if the real rank of G is zero or one. Hence for
connected simple Lie groups the weak Haagerup property coincides with weak
amenability. Moreover, it turns out that for connected simple Lie groups the
weak Haagerup constant coincides with the weak amenability constant, although
this is not true for locally compact groups in general.
It is also shown that the semidirect product of R^2 by SL(2,R) does not have
the weak Haagerup property.Comment: 19 pages. Final version. To appear in IMR
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