Noncommutative Analysis

Category: 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}.

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.

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.

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

Michael Skeide and I have recently uploaded our new paper to the arxiv: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond. In this gigantic (219 pages) paper, we propose a framework for studying the dilation theory of CP-semigroups parametrized by rather general monoids (i.e., semigroups with unit), and we use this framework for obtaining new results regarding the possibility or impossibility of constructing or having a dilation, we use it also for obtaining new structural results on the “mechanics” of dilations, and we analyze many examples using our tools. We present results that we have announced long ago, as well as some surprising discoveries.

This is an exciting moment for me, since we have been working on this project for more than a decade.

“Excuse me, did you really say decade?”

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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 🙂