Searcharxiv⌕ Search

arXiv subjects

Dan-Virgil Voiculescu

Publications and source records attributed to Dan-Virgil Voiculescu.

At least 19 recordsLinked to original sources
arXiv2025

Perturbations of operators and non-commutative condensers, an update on the quasicentral modulus

This is an update on the quasicentral modulus, an invariant for an n-tuple of Hilbert space operators and a rearrangement invariant norm, that plays a key-role in sharp multivariable generalizations of the classical Weyl-von Neumann-Kuroda and Kato-Rosenblum theorems of perturbation theory. There are also connections with self-similar measures on certain fractals and to the Kolmogorov-Sinai dynamical entropy. Some open problems are also pointed out. Recently a non-commutative analogy with condenser capacity in nonlinear potential theory is emerging, that provides a new perspective on the subject.

math.FA↗
arXiv2022

The condenser quasicentral modulus

We introduced the quasicentral modulus to study normed ideal perturbations of operators. It is a limit of condenser quasicentral moduli in view of a recently noticed analogy with capacity in nonlinear potential theory. We prove here some basic properties of the condenser quasicentral modulus and compute a simple example. We also discuss some associated variational problems. Part of the results are in the more general setting of a semifinite von Neumann algebra.

math.FA↗
arXiv2022

Capacity and the quasicentral modulus

We point out that the quasicentral modulus is a noncommutative analogue of a nonlinear rearrangement invariant Sobolev condenser capacity. In the case of the shifts by the generators of a finitely generated group, the quasicentral modulus coincides with a corresponding nonlinear condenser capacity on the Cayley graph of the group. Some other capacities related to the quasicentral modulus are also discussed.

math.FA↗
arXiv2021

Miscellaneous on Commutants mod Normed Ideals and Quasicentral Modulus I

We define commutants mod normed ideals associated with compact smooth manifolds with boundary. The results about the K-theory of these operator algebras include an exact sequence for the connected sum of manifolds, derived from the Mayer-Vietoris sequence. We also make a few remarks about bicommutants mod normed ideals and about the quasicentral modulus for the quasinormed p-Schatten-von Neumann classes 0 < p < 1.

math.FA↗
arXiv2020

The Formula for the Quasicentral Modulus in the Case of Spectral Measures on Fractals

We prove a general ampliation homogeneity result for the quasicentral modulus of an n-tuple of operators with respect to the (p,1) Lorentz normed ideal. We use this to prove a formula involving Hausdorff measure for the quasicentral modulus of n-tuples of commuting Hermitian operators the spectrum of which is contained in certain Cantor-like self-similar fractals.

math.FA↗
arXiv2019

Some Results and a K-theory Problem about Threshold Commutants mod Normed Ideals

We extend to the case of a threshold ideal our result with J. Bourgain about the essential centre of the commutant mod a diagonalization ideal for a n-tuple of commuting Hermitian operators . We also compute the $K_0$-group of the commutant mod trace-class of a unitary operator with spectrum equal to its essential spectrum. We present the problem of computing the $K_1$-group for a commutant mod trace-class in its simplest case.

math.FA↗
arXiv2019

Commutants mod Normed Ideals

We survey the operator algebras arising as commutants modulo normed ideals of finite sets of hermitian operators and connections to perturbations of operators and noncommutative geometry.

math.OA↗
arXiv2019

A Hydrodynamic Exercise in Free Probability: Setting up Free Euler Equations

For the free probability analogue of Euclidean space endowed with the Gaussian measure we apply the approach of Arnold to derive Euler equations for a Lie algebra of non-commutative vector fields which preserve a certain trace. We extend the equations to vector fields satisfying non-commutative smoothness requirements. We introduce a cyclic vorticity and show that it satisfies a vorticity equation and that it produces a family of conserved quantities.

math.OA↗
arXiv2019

Hybrid normed ideal perturbations of n-tuples of operators II: weak wave operators

We prove a general weak existence theorem for wave operators for hybrid normed ideal perturbations. We then use this result to prove the invariance of Lebesgue absolutely continuous parts of n-tuples of commuting hermitian operators under hybrid normed ideal perturbations from a class studied in the first paper of this series.

math.FA↗
arXiv2018

Boolean Extremes and Dagum Distributions

We study the max-convolution and max-stable laws for Boolean independence and prove that these are Dagum distributions (also known as log-logistical distributions).

math.FA↗
arXiv2017

Hybrid Normed Ideal Perturbations of n-tuples of Operators I

In hybrid normed ideal perturbations of $n$-tuples of operators, the normed ideal is allowed to vary with the component operators. We begin extending to this setting the machinery we developed for normed ideal perturbations based on the modulus of quasicentral approximation and an adaptation of our non-commutative generalization of the Weyl--von~Neumann theorem. For commuting $n$-tuples of hermitian operators, the modulus of quasicentral approximation remains essentially the same when $\cC_n^-$ is replaced by a hybrid $n$-tuple $\cC_{p_1,\dots}^-,\dots,\cC^-_{p_n}$, $p_1^{-1} + \dots + p_n^{-1} = 1$. The proof involves singular integrals of mixed homogeneity.

math.OA↗
arXiv2017

A remark about supramenability and the Macaev norm

We show that a finitely generated group G which satisfies a certain condition with respect to the Macaev norm is supramenable. The condition is equivalent to the existence of quasicentral approximate unit with respect to the Macaev norm relative to the left regular representation of the group and has been studied by the author in connection with perturbation questions for Hilbert space operators. The condition can be also viewed as an analogue with respect to the Macaev norm of Yamasaki parabolicity. We also show that existence of quasicentral approximate units relative to the Macaev norm for n-tuples of operators is not preserved when taking tensor products.

math.GR↗
arXiv2016

Lebesgue decomposition of functionals and unique preduals for commutants modulo normed ideals

We prove an analogue of the Lebesgue decomposition for continuous functionals on the commutant modulo a reflexive normed ideal of an n-tuple of hermitian operators for which there are quasicentral approximate units relative to the normed ideal. Using results of Godefroy-Talagrand and Pfitzner we derive from this strong uniqueness of the predual of such a commutant modulo a normed ideal.

math.OA↗
arXiv2016

K-theory and perturbations of absolutely continuous spectra

We study the K_0 group of the commutant modulo a normed ideal of an n-tuple of commuting Hermitian operators in some of the simplest cases. In case n=1, the results, under some technical conditions are rather complete and show the key role of the absolutely continuous part when the ideal is the trace-class. For a commuting n-tuple, n>2 and the Lorentz (n, 1) ideal, we show under an absolute continuity assumption that the commutant determines a canonical direct summand in K_0. Also, certain properties involving the compact ideal, established assuming quasicentral approximate units mod the normed ideal, have weaker versions which hold assuming only finiteness of the obstruction to quasicentral approximate units.

math.KT↗
arXiv2015

Free Probability for Pairs of Faces IV: Bi-free Extremes in the Plane

We compute the bi-free max-convolution which is the operation on bi-variate distribution functions corresponding to the max-operation with respect to the spectral order on bi-free bi-partite two-faced pairs of hermitian non-commutative random variables. With the corresponding definitions of bi-free max-stable and max-infinitely-divisible laws their determination becomes in this way a classical analysis question.

math.OA↗
arXiv2013

Countable Degree-1 Saturation of Certain C*-Algebras Which Are Coronas of Banach Algebras

We study commutants modulo some normed ideal of n-tuples of operators which satisfy a certain approximate unit condition relative to the ideal. We obtain results about the quotient of these Banach algebras by their ideal of compact operators being C*-algebras which have the countable degree-1 saturation property in the model-theory sense of I. Farah and B. Hart. We also obtain results about quasicentral approximate units, multipliers and duality.

math.OA↗