Boundary representations of operator spaces, and compact rectangular matrix convex sets

TL;DR

This paper develops a new framework for matrix convexity in operator spaces, establishing fundamental theorems, a correspondence with operator spaces, and a boundary representation theory that generalizes Arveson's conjecture.

Contribution

It introduces compact rectangular matrix convex sets, proves analogs of classical theorems, and establishes boundary representations for operator spaces, including a canonical construction of the triple envelope.

Findings

Established a correspondence between convex sets and operator spaces.

Proved the analog of Arveson's conjecture for boundary representations.

Developed a framework for compact rectangular matrix convex sets.

Abstract

We initiate the study of matrix convexity for operator spaces. We define the notion of compact rectangular matrix convex set, and prove the natural analogs of the Krein-Milman and the bipolar theorems in this context. We deduce a canonical correspondence between compact rectangular matrix convex sets and operator spaces. We also introduce the notion of boundary representation for an operator space, and prove the natural analog of Arveson's conjecture: every operator space is completely normed by its boundary representations. This yields a canonical construction of the triple envelope of an operator space.