## Category: Operator algebras

### Slides of my talk at the seminar “in” Bucharest

This Wednesday I gave a talk at the Institute of Mathematics in Bucharest, live on zoom. Here are the slides:

In this talk, I decided to put an emphasis on telling the story of how we found ourselves working on this problem, rather than giving a logical presentation of the results in the paper that I was trying to advertise (this paper). I am not sure how much of this story one can get from the slides, but here they are.

### Seminar talk by Dor-On: Quantum symmetries in the representation theory of operator algebras

NOTE: THE SEMINAR WAS POSTPONED TO DECEMBER 10.

On next Thursday the Operator Algebras and Operator Seminar will convene for a talk by Adam Dor-On.

Title: Quantum symmetries in the representation theory of operator algebras

Speaker: Adam Dor-On (University of Illinois, Urbana-Champaign)

Time: AFTERNOON Thursday Dec. 10, 2020 (NOTE: THE SEMINAR WAS POSTPONED BY ONE WEEK FROM ORIGINAL DATE).

(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Abstract:

We introduce a non-self-adjoint generalization of Quigg’s notion of coaction of a discrete group G on a C*-algebra. We call these coactions “quantum symmetries” because from the point of view of quantum groups, coactions on C*-algebras are just actions of a quantum dual group of G on the C*-algebra. We introduce and develop a compatible C*-envelope, which is the smallest C*-coaction system which contains a given operator algebra coaction system, and we call it the cosystem C*-envelope.

It turns out that the new point of view of quantum symmetries of non-self-adjoint algebras is useful for resolving problems in both C*-algebra theory and non-self-adjoint operator algebra theory. We use quantum symmetries to resolve some problems left open in work of Clouatre and Ramsey on finite dimensional approximations of representations, as well as a problem of Carlsen, Larsen, Sims and Vitadello on the existence of a co-universal C*-algebra for product systems over arbitrary right LCM semigroup embedded in groups. This latter problem was resolved for abelian lattice ordered semigroups by the speaker and Katsoulis, and we extend this to arbitrary right LCM semigroups. Consequently, we are also able to extend the Hao-Ng isomorphism theorems of the speaker with Katsoulis from abelian lattice ordered semigroups to arbitrary right LCM semigroups.

*This talk is based on two papers. One with Clouatre, and another with Kakariadis, Katsoulis, Laca and X. Li.

### Seminar talk by Salomon: Combinatorial and operator algebraic aspects of proximal actions

The Operator Algebras and Operator Theory Seminar is back (sort of). This semester we will have the seminar on Thursdays 15:30 (Israel time) about once in a while. Please send me an email if you want to join the mailing list and get the link for the zoom meetings. Here are the details for our first talk :

Title: Combinatorial and operator algebraic aspects of proximal actions

Speaker: Guy Salomon (Weizmann Institute)

Time: 15:30-16:30,Thursday Nov. 12, 2020

(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Abstract:

An action of a discrete group $G$ on a compact Hausdorff space $X$ is called proximal if for every two points $x$ and $y$ of $X$ there is a net $g_i \in G$ such that $\lim(g_i x)=\lim(g_i y)$, and strongly proximal if the natural action of $G$ on the space $P(X)$ of probability measures on $X$ is proximal. The group $G$ is called strongly amenable if all of its proximal actions have a fixed point and amenable if all of its strongly proximal actions have a fixed point.

In this talk I will present relations between some fundamental operator theoretic concepts to proximal and strongly proximal actions, and hence to amenable and strongly amenable groups. In particular, I will focus on the C*-algebra of continuous functions over the universal minimal proximal $G$-flow and characterize it in the category of $G$-operator-systems.

I will then show that nontrivial proximal actions of $G$ can arise from partitions of $G$ into a certain kind of “large” subsets. If time allows, I will also present some relations to the Poisson boundaries of $G$. The talk is based on a joint work with Matthew Kennedy and Sven Raum.

### 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 unitary tuples

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 $c({\mathrm u},u_0)\geq\sqrt{3}$.