## Tag: dilation theory

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

### A survey (another one!) on dilation theory

I recently uploaded to the arxiv my new survey “Dilation theory: a guided tour“. I am pretty happy and proud of the result! Right now I feel like it is the best survey ever written (honest, that’s how I feel, I know that its an illusion), but experience tells me that two months from now I might be a little embarrassed (like: how could I be so vain to think that I could pull of a survey on this gigantic topic?).

(Well, these are the usual highs and lows of being a mathematician, but since this is a survey paper and not a research paper, I feel comfortable enough to share these feelings).

This survey was submitted (and will hopefully appear in) to the Proceedings of the International Workshop on Operator Theory and its Applications (IWOTA) 2019, Portugal. It is an expanded version of the semi-plenary talk that I gave in that conference. I used a preliminary version of this survey as lecture notes for the mini-course that I gave at the recent workshop “Noncommutative Geometry and its Applications” at NISER Bhubaneswar.

I hope somebody finds it useful or entertaining 🙂

### My talks at NCGA Bhubaneswar

This week I am giving a series of five lectures on dilation theory in the workshop Noncommutative Geometry and its Applications , at NISER Bhubaneswar (India). I am putting my talks up here in case anybody would like to see them (and also as backup, in case my stick doesn’t work).

(Lectures 1,2,3 were board talks).

### The complex matrix cube problem – new results from summer projects

In this post I will summarize the results obtained by my group in the “Summer Projects Week” that took place two weeks ago at the Technion. As in last time (see here for a summary of last year’s project) the title of the project I suggested was “Numerical Explorations of Open Problems from Operator Theory”. This time, I was lucky to have Malte Gerhold and Satish Pandey, my postdocs, volunteer to help me with the mentoring. The students who chose our project were Matan Gibson and Ofer Israelov, and they did fantastic work.

Read the rest of this entry »

### The complex matrix cube problem summer project – summary of results

In the previous post I announced the project that I was going to supervise in the Summer Projects in Mathematics week at the Technion. In this post I wish to share what we did and what we found in that week.

I had the privilege to work with two very bright students who have recently finished their undergraduate studies: Mattya Ben-Efraim (from Bar-Ilan University) and Yuval Yifrach (from the Technion). It is remarkable the amount of stuff they learned for this one week project (the basics of C*-algebras and operator spaces), and that they actually helped settle the question that I raised to them.

I learned a lot of things in this project. First, I learned that my conjecture was false! I also learned and re-learned some programming abilities, and I learned something about the subtleties and limitations of numerical experimentation (I also learned something about how to supervise an undergraduate research project, but that’s besides the point right now).