721 research outputs found

    A computational approach to the Thompson group FF

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    Let FF denote the Thompson group with standard generators A=x0A=x_0, B=x1B=x_1. It is a long standing open problem whether FF is an amenable group. By a result of Kesten from 1959, amenability of FF is equivalent to (i)∣∣I+A+B∣∣=3(i)\qquad ||I+A+B||=3 and to (ii)∣∣A+A−1+B+B−1∣∣=4,(ii)\qquad ||A+A^{-1}+B+B^{-1}||=4, where in both cases the norm of an element in the group ring CF\mathbb{C} F is computed in B(ℓ2(F))B(\ell^2(F)) via the regular representation of FF. By extensive numerical computations, we obtain precise lower bounds for the norms in (i)(i) and (ii)(ii), as well as good estimates of the spectral distributions of (I+A+B)∗(I+A+B)(I+A+B)^*(I+A+B) and of A+A−1+B+B−1A+A^{-1}+B+B^{-1} with respect to the tracial state τ\tau on the group von Neumann Algebra L(F)L(F). Our computational results suggest, that ∣∣I+A+B∣∣≈2.95∣∣A+A−1+B+B−1∣∣≈3.87.||I+A+B||\approx 2.95 \qquad ||A+A^{-1}+B+B^{-1}||\approx 3.87. It is however hard to obtain precise upper bounds for the norms, and our methods cannot be used to prove non-amenability of FF.Comment: appears in International Journal of Algebra and Computation (2015

    Quasitraces on exact C*-algebras are traces

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    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

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    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

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    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

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    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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