Noncommutative Analysis

Category: Operator algebras

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}.

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.

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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