Boundary Representations and Rectangular Hyperrigidity

TL;DR

This paper investigates boundary representations of operator spaces, introduces the concept of rectangular hyperrigidity, and establishes finite-dimensional analogues of classical theorems, advancing the understanding of operator space structure.

Contribution

It introduces rectangular hyperrigidity and characterizes boundary representations for operator spaces, extending classical results to the finite-dimensional operator space setting.

Findings

Boundary representations characterized via finite representation and separating property.

Introduction of weak boundary for operator spaces.

Establishment of a finite-dimensional analogue of Saskin's theorem.

Abstract

We explore connections between boundary representations of operator spaces and those of the associated Paulsen systems. Using the notions of finite representation and separating property which we introduced, boundary representations for operator spaces is characterised. We also introduce weak boundary for operator spaces. Rectangular hyperrigidity of operator spaces introduced here is used to establish an analogue of Saskin's theorem in the setting of operator spaces in finite dimensions.