Noncommutative Analysis

Month: June, 2020

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}.