arXiv subjects

Uffe Haagerup

Publications and source records attributed to Uffe Haagerup.

41 records · Page 3Linked to original sources

arXiv2003

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.

math.OA↗

arXiv2002

Random matrices, free probability and the invariant subspace problem relative to a von Neumann Algebra

Random matrices have their roots in multivariate analysis in statistics, and since Wigner's pioneering work in 1955, they have been a very important tool in mathematical physics. In functional analysis, random matrices and random structures have in the last two decades been used to construct Banach spaces with surprising properties. After Voiculescu in 1990--1991 used random matrices to classification problems for von Neumann algebras, they have played a key role in von Neumann algebra theory. In this lecture we will discuss some new applications of random matrices to operator algebra theory, namely applications to classification problems for $C^*$-algebras and to the invariant subspace problem relative to a von Neumann algebra.

math.OA↗

arXiv2002

DT-operators and decomposability of Voiculescu's circular operator

The DT-operators are introduced, one for every pair (μ,c) consisting of a compactly supported Borel probability measure μon the complex plane and a constant c>0. These are operators on Hilbert space that are defined as limits in *-moments of certain upper triangular random matrices. The DT-operators include Voiculescu's circular operator and elliptic deformations of it, as well as the circular free Poisson operators. We show that every DT-operator is strongly decomposable. We also show that a DT-operator generates a II_1-factor, whose isomorphism class depends only on the number and sizes of atoms of μ. Those DT-operators that are also R-diagonal are identified. For a quasi-nilpotent DT-operator T, we find the distribution of T^*T and a recursion formula for general *-moments of T.

math.OA↗

arXiv2000

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.

math.OA↗

arXiv1996

The stable rank of some free product C*-algebras

It is proved that the reduced group C*-algebra C*_{red}(G) has stable rank one (i.e. its group of invertible elements is a dense subset) if G is a discrete group arising as a free product G_1*G_2 where |G_1|>=2 and |G_2|>=3. This follows from a more general result where it is proved that if (A,tau) is the reduced free product of a family (A_i,tau_i), i\in I, of unital C*-algebras A_i with normalized faithful traces tau_i, and if the family satisfies the Avitzour condition (i.e. the traces, tau_i, are not too lumpy in a specific sense), then A has stable rank one.

funct-an↗