Ternary rings of operators and their linking von Neumann algebras

Abstract

In this short note, we show that a von Neumann algebra can be written as the linking von Neumann algebra of a $W^\ast$-ternary ring of operators ($W^\ast$-TRO, in short), if and only if, it contains no abelian direct summand. We also provide some new characterizations for nuclear TROs and $W^\ast$-exact TROs.