## Category: Operator algebras

### Seminar talk at the BGU OA Seminar

This coming Thursday (July 2nd, 14:10 Israel Time) I will be giving a talk at the Ben-Gurion University Math Department’s Operator Algebras Seminar. If you are interested in a link to the Zoom please send me an email.

I will be talking mostly about these two papers of mine with co-authors: older one, newer one. Here is the title and abstract:

Title: Matrix ranges, fields, dilations and representations

Abstract: In my talk I will present several results whose unifying theme is a matrix-valued analogue of the numerical range, called the matrix range of an operator tuple. After explaining what is the matrix range and what it is good for, I will report on recent work in which we prove that there is a certain “universal” matrix range, to which the matrix ranges of a sequence of large random matrices tends to, almost surely. The key novel technical aspects of this work are the (levelwise) continuity of the matrix range of a continuous field of operators, and a certain quantitative matrix valued Hahn-Banach type separation theorem. In the last part of the talk I will explain how the (uniform) distance between matrix ranges can be interpreted equivalently as a “dilation distance”, which can be interpreted as a kind of “representation distance”. These vague ideas will be illustrated with an application: the construction of a norm continuous family of representations of the noncommutative tori (recovering a result of Haagerup-Rordam in the d=2 case and of Li Gao in the d>2 case).

Based on joint works with Malte Gerhold, Satish Pandey and Baruch Solel.

### New paper: Dilations of commuting unitaries

Malte Gerhold, Satish Pandey, Baruch Solel and I have recently posted a new paper on the arxiv. Check it out here. Here is the abstract:

Abstract:

We study the space of all $d$-tuples of unitaries $u=(u_1,\ldots, u_d)$ using dilation theory and matrix ranges. Given two $d$-tuples $u$ and $v$ generating C*-algebras $\mathcal A$ and $\mathcal B$, we seek the minimal dilation constant $c=c(u,v)$ such that $u\prec cv$, by which we mean that $u$ is a compression of some $*$-isomorphic copy of $cv$. This gives rise to a metric

$d_D(u,v)=\log\max\{c(u,v),c(v,u)\}$

on the set of equivalence classes of $*$-isomorphic tuples of unitaries. We also consider the metric

$d_{HR}(u,v)$ $= \inf \{\|u'-v'\|:u',v'\in B(H)^d, u'\sim u$ and $v'\sim v\},$

and we show the inequality

$d_{HR}(u,v) \leq K d_D(u,v)^{1/2}.$

Let $u_\Theta$ be the universal unitary tuple $(u_1,\ldots,u_d)$ satisfying $u_\ell u_k=e^{i\theta_{k,\ell}} u_k u_\ell$, where $\Theta=(\theta_{k,\ell})$ is a real antisymmetric matrix. We find that $c(u_\Theta, u_{\Theta'})\leq e^{\frac{1}{4}\|\Theta-\Theta'\|}$. From this we recover the result of Haagerup-Rordam and Gao that there exists a map $\Theta\mapsto U(\Theta)\in B(H)^d$ such that $U(\Theta)\sim u_\Theta$ and

$\|U(\Theta)-U({\Theta'})\|\leq K\|\Theta-\Theta'\|^{1/2}.$

Of special interest are: the universal $d$-tuple of noncommuting unitaries ${\mathrm u}$, the $d$-tuple of free Haar unitaries $u_f$, and the universal $d$-tuple of commuting unitaries $u_0$. We obtain the bounds

$2\sqrt{1-\frac{1}{d}}\leq c(u_f,u_0)\leq 2\sqrt{1-\frac{1}{2d}}.$

From this, we recover Passer’s upper bound for the universal unitaries $c({\mathrm u},u_0)\leq\sqrt{2d}$. In the case $d=3$ we obtain the new lower bound $c({\mathrm u},u_0)\geq 1.858$ improving on the previously known lower bound $c({\mathrm u},u_0)\geq\sqrt{3}$.

### My slides for the COSY talk and the seminar talk

Here is a link to the slides for the short talk that I am giving in COSY.

This talk is a short version of the talk I gave at the Besancon Functional Analysis Seminar last week; here are the slides for that talk.

### Seminar talk

Next Tuesday, May 19th, at 14:30 (Israeli time), I will give a video talk at the Séminaire d’Analyse Fonctionnelle “in” Laboratoire de mathématiques de Besançon. It will be about my recent paper with Michael Skeide, the one that I announced here.

Title: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: the Multi-Parameter Case and Beyond.

Abstract: We introduce a framework for studying dilations of semigroups of completely positive maps on von Neumann algebras. The heart of our method is the systematic use of families of Hilbert C*-correspondences that behave nicely with respect to tensor products: these are product systems, subproduct systems and superproduct systems. Although we developed our tools with the goal of understanding the multi-parameter case, they also lead to new results even in the well studied one parameter case. In my talk I will give a broad outline and a taste of the dividends our work.

The talk is based on a recent joint work with Michael Skeide.

Assumed knowledge: Completely positive maps and C*-algebras.

Feel free to write to me if you are interested in a link to the video talk.

### Dilations of q-commuting unitaries

Malte Gerhold and I just have just uploaded a revision of our paper “Dilations of q-commuting unitaries” to the arxiv. This paper has been recently accepted to appear in IMRN, and was previously rejected by CMP, so we have four anonymous referees and two handling editors to be thankful to for various corrections and suggested improvements (though, as you may understand, one editor and two referees have reached quite a wrong conclusion regarding our beautiful paper :-).

This is a quite short paper (200 full pages shorter than the paper I recently announced), which tells a simple and interesting story: we find that optimal constant $c_\theta$, such that every pair of unitaries $u,v$ satisfying the q-commutation relation

$vu = e^{i\theta} uv$

dilates to a pair of commuting normal operators with norm less than or equal to $c_\theta$ (this problems is related to the “complex matrix cube problem” that we considered in the summer project half year ago and the one before). We provide a full solution. There are a few ramifications of this idea, as well as surprising connections and applications, so I invite you to check out the nice little introduction.