Unital operator spaces and discrete groups
Nikolaos Koutsonikos-Kouloumpis
TL;DR
The paper addresses when a unital operator space has the trivial intersection property and how this controls its $C^*$-envelope, with an application to discrete groups. It develops criteria ensuring $C^*_e(\mathcal{U})\cong C^*(\mathcal{U})$ and proves that unital surjective complete isometries between spaces containing a discrete masa are unitary equivalences. It also analyzes operator spaces $\mathbb{M}(\Omega)$ arising from group actions and shows that $\mathbb{M}(E^\star)$ is a von Neumann algebra exactly when $E$ is a subgroup, linking algebraic and operator-algebraic structures. Altogether, the work connects trivial intersection properties, boundary-ideal phenomena, and $C^*$-envelopes to group-theoretic properties in a unified operator-space framework.
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 $\ell^2(Γ)$ for any set $Γ$, that contain the masa $\ell^\infty(Γ)$, 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 $G$.