Unital operator spaces and discrete groups

TL;DR

This paper studies unital operator spaces with the trivial intersection property, showing they lack nontrivial boundary ideals, and explores their structure, isometries, and connections to group properties.

Contribution

It introduces the trivial intersection property for operator spaces, characterizes spaces with this property, and links these spaces to algebraic properties of groups.

Findings

Spaces with the trivial intersection property have no nontrivial boundary ideals.

Unital operator spaces on ()) containing ()) possess the trivial intersection property.

Complete isometries between such spaces are unitary equivalences.

Abstract

We introduce the trivial intersection property for concrete operator spaces and we show that a unital space with this property has no nontrivial boundary ideals. We provide various examples of such spaces, among which are strongly reflexive masa bimodules and completely distributive CSL algebras. We show that unital operator spaces acting on $\ell^2(\Gamma)$ for any set $\Gamma$, that contain the masa $\ell^\infty(\Gamma)$, possess the trivial intersection property, and we use this to prove that a unital surjective complete isometry between such spaces is a unitary equivalence. Then, we apply these results to $w^*$-closed $\ell^\infty(G)$-bimodules acting on $\ell^2(G)$ for a group $G$ and we relate them to algebraic properties of $G$.