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Ken Dykema

Publications and source records attributed to Ken Dykema.

At least 19 recordsLinked to original sources
arXiv2025

Free products and rescalings involving non-separable abelian von Neumann algebras

For a self-symmetric tracial von Neumann algebra $A$, we study rescalings of $A^{*n} * L\mathbb{F}_r$ for $n \in \mathbb{N}$ and $r \in (1, \infty]$ and use them to obtain an interpolation $\mathcal{F}_{s,r}(A)$ for all real numbers $s>0$ and $1-s < r \leq \infty$. We get formulas for their free products, and free products with finite-dimensional or hyperfinite von Neumann algebras. In particular, for any such $A$, we can compute compressions $(A^{*n})^t$ for $0<t<1$, and the Murray-von Neumann fundamental group of $A^{*\infty}$. When $A$ is also non-separable and abelian, this answers two questions in Section 4.3 of recent work of Boutonnet-Drimbe-Ioana-Popa.

math.OA↗
arXiv2024

On operator valued Haar unitaries and bipolar decompositions of R-diagonal elements

In the context of operator valued W*-free probability theory, we study Haar unitaries, R-diagonal elements and circular elements. Several classes of Haar unitaries are differentiated from each other. The term bipolar decomposition is used for the expression of an element as $vx$ where $x$ is self-adjoint and $v$ is a partial isometry, and we study such decompositions of operator valued R-diagonal and circular elements that are free, meaning that $v$ and $x$ are *-free from each other. In particular, we prove, when B=C^2, that if a $B$-valued circular element has a free bipolar decomposition with $v$ unitary, then it has one where $v$ normalizes $B$.

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arXiv2023

On algebra-valued R-diagonal elements (with erratum)

For an element in an algebra-valued *-noncommutative probability space, equivalent conditions for algebra-valued R-diagonality (a notion introduced by Sniady and Speicher) are proved. Formal power series relations involving the moments and cumulants of such R-diagonal elements are proved. Decompositions of algebra-valued R-diagonal elements into products of the form unitary times self-adjoint are investigated; sufficient conditions, in terms of cumulants, for *-freeness of the unitary and the self-adjoint part are proved, and a tracial example is given where *-freeness fails. The particular case of algebra-valued circular elements is considered; as an application, the polar decompostion of the quasinilpotent DT-operator is described. An erratum is appended. (The last sentence in the above paragraph is thereby nullified.)

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arXiv2021

Some non-spectral DT-operators in finite von Neumann algebras

Given a DT-operator $Z$ whose Brown measure is radially symmetric and has a certain concentration property, it is shown that $Z$ is not spectral in the sense of Dunford. This is accomplished by showing that the angles between certain complementary Haagerup-Schultz projections of $Z$ are zero. New estimates on norms and traces of powers of algebra-valued circular operators over commutative C$^*$-algebras are also proved.

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arXiv2020

Angles between Haagerup--Schultz projections and spectrality of operators

We investigate angles between Haagerup--Schultz projections of operators belonging to finite von Neumann algebras, in connection with a property analogous to Dunford's notion of spectrality of operators. In particular, we show that an operator can be written as the sum of a normal and an s.o.t.-quasinilpotent operator that commute if and only if the angles between its Haagerup--Schultz projections are uniformly bounded away from zero (and we call this the uniformly nonzero anlges property). Moreover, we show that spectrality is equivalent to this uniformly nonzero angles property plus decomposability. Finally, using this characterization, we construct an easy example of an operator which is decomposable but not spectral, and we show that Voiculescu's circular operator is not spectral (nor are any of the circular free Poisson operators).

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arXiv2019

Simultaneous upper triangular forms for commuting operators in a finite von Neumann algebra

The joint Brown measure and joint Haagerup--Schultz projections for tuples of commuting operators in a von Neumann algebra equipped with a faithful tracial state are investigated, and several natural properties are proved for these. It is shown that the support of the joint Brown measure is contained in the Taylor joint spectrum of the tuple, and also in the ostensibly smaller left Harte spectrum. A simultaneous upper triangularization result for finite commuting tuples is proved and the joint Brown measure and joint Haagerup--Schultz projections are shown to be have well under the Arens multivariate holomorphic functional calculus of such a commuting tuple.

math.OA↗
arXiv2017

Decomposibility and norm convergence properties in finite von Neumann algebras

We study Schur-type upper triangular forms for elements, T, of von Neumann algebras equipped with faithful, normal, tracial states. These were introduced in a paper of Dykema, Sukochev and Zanin; they are based on Haagerup-Schultz projections. We investigate when the s.o.t.-quasinilpotent part of this decomposition of T is actually quasinilpotent. We prove implications involving decomposability and strong decomposability of T. We show this is related to norm convergence properties of the sequence |T^n|^{1/n} which, by a result of Haagerup and Schultz, is known to converge in strong operator topology. We introduce a Borel decomposability, which is a property appropriate for elements of finite von Neumann algebras, and show that the circular operator is Borel decomposable. We also prove the existence of a thin-spectrum s.o.t.-quasinilpotent operator in the hyperfinite II_1-factor.

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arXiv2017

The Delta Game

We introduce a game related to the $I_{3322}$ game and analyze a constrained value function for this game over various families of synchronous quantum probability densities.

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arXiv2017

Asymptotic *-moments of some random Vandermonde matrices

Appropriately normalized square random Vandermonde matrices based on independent random variables with uniform distribution on the unit circle are studied. It is shown that as the matrix sizes increases without bound, with respect to the expectation of the trace there is an asymptotic *-distribution, equal to that of a C[0,1]-valued R-diagonal element.

math.PR↗
arXiv2016

KMS quantum symmetric states

Let $A$ be a unital C$^*$-algebra and let $σ$ be a one-parameter automorphism group of $A$. We consider $\operatorname{QSS}_σ(A)$, the set of all quantum symmetric states on $*_1^\infty A$ that are also KMS states (for a fixed inverse temperature, for specificity taken to be $-1$) for the free product automorphism group $*_1^\inftyσ$. We characterize the elements of $\operatorname{QSS}_σ(A)$, we show that $\operatorname{QSS}_σ(A)$ is a Choquet simplex whenever it is nonempty and we characterize its extreme points.

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arXiv2016

Canonical random variables for multivariate, algebra-valued distributions

In the algebraic theory of algebra-valued noncommutative probability spaces, for a unital algebra B, a mild reformulation of Speicher's noncrossing B-valued cumulants for random variables in these spaces is used to construct canonical random variables, acting on a Fock space over B, for arbitrary families of B-valued random variables. Also, a condition for traciality of a trace on B composed with the B-valued conditional expectation is given in terms of B-valued cumulants.

math.RA↗
arXiv2015

Numerical Ranges in II$_1$ Factors

In this paper, we generalize the notion of the $C$-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator $C$, the $C$-numerical range of such an operator is defined; it is a compact, convex subset of $\mathbb{C}$. We explicitly describe the $C$-numerical ranges of several operators and classes of operators.

math.OA↗
arXiv2015

An upper triangular decomposition theorem for some unbounded operators affiliated to II_1-factors

Results of Haagerup and Schultz (2009) about existence of invariant subspaces that decompose the Brown measure are extended to a large class of unbounded operators affiliated to a tracial von Neumann algebra. These subspaces are used to decompose an arbitrary operator in this class into the sum of a normal operator and a spectrally negligible operator. This latter result is used to prove that, on a bimodule over a tracial von Neumann algebra that is closed with respect to logarithmic submajorization, every trace is spectral, in the sense that the trace value on an operator depends only on the Brown measure of the operator.

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arXiv2015

Algebras of log-integrable functions and operators

We show that certain spaces of log-integrable functions and operators are complete topological *-algebras with respect to a natural metric space structure. We explore connections with the Nevanlinna class of holomorphic functions.

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arXiv2015

On reduction theory and Brown measure for closed unbounded operators

The theory of direct integral decompositions of both bounded and unbounded operators is further developed; in particular, results about spectral projections, functional calculus and affiliation to von Neumann algebras are proved. For operators belonging to or affiliated to a tracial von Neumann algebra that is a direct integral von Neumann algebra, the Brown measure is shown to be given by the corresponding integral of Brown measures.

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