TRO equivalent algebras

TL;DR

This paper introduces TRO equivalence as a new relation between operator algebras, characterizes it for reflexive algebras, and explores its implications for CSL algebras and their properties.

Contribution

It defines TRO equivalence for w*-closed operator algebras, characterizes it for reflexive algebras, and compares it with spatial Morita equivalence, revealing new structural insights.

Findings

TRO equivalence is characterized by isomorphisms of commutants of diagonals.

TRO equivalence is stronger than spatial Morita equivalence.

TRO equivalence preserves properties like syntheticity in CSL algebras.

Abstract

In this work we study a new equivalence relation between w* closed algebras of operators on Hilbert spaces. The algebras A and B are called TRO equivalent if there exists a ternary ring of operators M (i.e. MM*M\subset M) such that A is the w*-closed span of M*BM and B is the w*-closed span of MAM*. We prove that two reflexive algebras are TRO equivalent if and only if there exists a * isomorphism between the commutants of their diagonals mapping the invariant projection lattice of the first algebra onto the lattice of the second one. We explore some consequences of TRO equivalence for CSL algebras. We also prove that TRO equivalence is stronger than "spatial Morita equivalence". Two CSL algebras are "spatially Morita equivalent" if and only if their lattices are isomorphic. In this case if one of them is synthetic then so is the other.