The linking von Neumann algebras of W*-TROs

TL;DR

This paper characterizes when a von Neumann algebra can be represented as a linking von Neumann algebra of a W*-TRO, linking algebraic properties with structural conditions.

Contribution

It provides a necessary and sufficient condition for a von Neumann algebra to be a linking algebra of a W*-TRO and introduces new characterizations of nuclear and W*-exact TROs.

Findings

A von Neumann algebra is a linking von Neumann algebra of a W*-TRO iff it has no abelian direct summand.

New characterizations of nuclear TROs in terms of their linking algebras.

New characterizations of W*-exact TROs based on linking algebra properties.

Abstract

In this note, we show that a von Neumann algebra can be written as the linking von Neumann algebra of a W*-TRO if and only if it contains no abelian direct summand. We also provide some new characterizations of nuclear TROs and $W^\ast$-exact TROs in terms of the properties of their linking algebras.