Noncommutative Analysis

Category: Operator 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}.

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.

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 🙂

New paper: “On the matrix range of random matrices”

Malte Gerhold and I recently posted our new paper “On the matrix range of random matrices” on the arxiv, and I want to write a few words about it.

Recall that the matrix range of a d-tuple of operators A = (A_1, \ldots, A_d) \in B(H)^d is the noncommutative set W(A) = \cup_n W_n(A), where

W_n(A) = \{ (\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n is UCP \}.

The matrix range appeared in several recent papers of mine (for example this one), it is a complete invariant for the unital operator space generated by A_1 \ldots, A_d, and is within some classes is also a unitary invariant.

The idea for this paper came from my recent (last couple of years or so) flirt with numerical experiments. It has dawned on me that choosing matrices randomly from some ensembles, for example by setting

G = randn(N);

X = (G + G')/sqrt(2*N);

(this is the GOE ensemble) is a rather bad way for testing “classical” conjectures in mathematics, such as what is the best constant for some inequality. Rather, as N increases, random N \times N behave in a very “structured” way (as least in some sense). So we were driven to try to understand, roughly what kind of operator theoretic phenomena do we tend to observe when choosing random matrices.

The above paragraph is a confession of the origin of our motive, but at the end of the day we ask and answer honest mathematical questions with theorems and proofs. If X^N = (X^N_1, \ldots, X^N_d) is a d-tuple of N \times N matrices picked at random according to the Matlab code above, then experience with the law of large numbers, the central limit theorem, and Wigner’s semicircle law, suggests that W(X^N) will “converge” to something. And by experience with free probability theory, if it converges to something, then is should be the matrix range of the free semicircular tuple. We find that this is indeed what happens.

Theorem: Let X^N be as above, and let s = (s_1, \ldots, s_d) be a semicircular family. Then for all n,

\lim_{N \to \infty} d_H(W_n(X^N),W(s)) = 0 almost surely

in the Hausdorff metric.

The semicircular tuple s = (s_1, \ldots, s_d) is a certain d-tuple of operators that can be explicitly described (see our paper, for example).

We make heavy use of some fantastic results in free probability and random matrix theory, and our contribution boils down to finding the way to use existing results in order to understand what happens at the level of matrix ranges. This involves studying the continuity of matrix ranges for continuous fields of operators, in particular, we study the relationship between the convergence

(*) \lim_{N \to \infty} \|p(X^N)\| = \|p(X)\|

(which holds for X^N as above and X = s by a result of Haagerup and Torbjornsen) and

(**) \lim_{N \to \infty} d_H(W_n(X^N),W(X)) = 0.

To move from (*) to (**), we needed to devise a certain quantitative Effros-Winkler Hahn-banach type separation theorem for matrix convex sets.

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.

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