# Unperforated pairs of operator spaces and hyperrigidity of operator systems

## TL;DR

This paper investigates the properties of unperforated pairs of operator spaces and their relation to hyperrigidity in operator systems, providing evidence for Arveson's conjecture and exploring the role of the weak expectation property.

## Contribution

It introduces the concept of unperforated pairs in operator spaces, proves their relation to hyperrigidity, and links the weak expectation property to unperforated pairs.

## Key findings

- Commuting pairs are unperforated.
- Unperforated pairs relate to hyperrigidity.
- Weak expectation property relaxes unperforated pair conditions.

## Abstract

We study restriction and extension properties for states on C$^*$-algebras with an eye towards hyperrigidity of operator systems. We use these ideas to provide supporting evidence for Arveson's hyperrigidity conjecture. Prompted by various characterizations of hyperrigidity in terms of states, we examine unperforated pairs of self-adjoint subspaces in a C$^*$-algebra. The configuration of the subspaces forming an unperforated pair is in some sense compatible with the order structure of the ambient C$^*$-algebra. We prove that commuting pairs are unperforated, and obtain consequences for hyperrigidity. Finally, by exploiting recent advances in the tensor theory of operator systems, we show how the weak expectation property can serve as a flexible relaxation of the notion of unperforated pairs.