Random Matrices and Subexponential Operator Spaces

TL;DR

This paper introduces subexponential operator spaces, extends Grothendieck-type factorization results to them using random matrices, and provides examples of non-exact subexponential spaces and a specific subexponential $C^*$-algebra.

Contribution

It generalizes the concept of exact operator spaces to subexponential ones and extends key factorization results, providing new examples and counterexamples.

Findings

Grothendieck-type factorization extends to subexponential spaces

Existence of non-exact subexponential operator spaces

Identification of certain operator spaces that are not subexponential

Abstract

We introduce and study a generalization of the notion of exact operator space that we call subexponential. Using Random Matrices we show that the factorization results of Grothendieck type that are known in the exact case all extend to the subexponential case, but we exhibit (a continuum of distinct) examples of non-exact subexponential operator spaces, as well as a $C^*$-algebra that is subexponential with constant 1 but not exact. We also show that $OH$, $R+C$ and $\max(\ell_2)$ (or any other maximal operator space) are not subexponential.