Recall that the matrix range of a -tuple of operators is the noncommutative set , where
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 , 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 increases, random 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 is a -tuple of 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 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 be as above, and let be a semicircular family. Then for all ,
in the Hausdorff metric.
The semicircular tuple is a certain -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
(which holds for as above and by a result of Haagerup and Torbjornsen) and
To move from (*) to (**), we needed to devise a certain quantitative Effros-Winkler Hahn-banach type separation theorem for matrix convex sets.