Universally reversible JC*-triples and operator spaces

TL;DR

This paper characterizes universally reversible JC*-triples by their inability to map onto large Hilbert spaces or spin factors, and explores their operator space structures and related properties.

Contribution

It provides a complete characterization of universally reversible JC*-triples and their operator space structures, linking algebraic and geometric properties.

Findings

Most JC*-triples are universally reversible.

Universal reversibility is characterized by absence of certain homomorphisms.

Unique operator space structure occurs when no nonabelian TRO ideals are present.

Abstract

We prove that the vast majority of JC*-triples satisfy the condition of universal reversibility. Our characterisation is that a JC*-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of dimension greater than two nor onto spin factors of dimension greater than four. We establish corresponding characterisations in the cases of JW*-triples and of TROs (regarded as JC*-triples). We show that the distinct natural operator space structures on a universally reversible JC*-triple E are in bijective correspondence with a distinguished class of ideals in its universal TRO, identify the Shilov boundaries of these operator spaces and prove that E has a unique natural operator space structure precisely when E contains no ideal isometric to a nonabelian TRO. We deduce some decomposition and completely contractive properties of triple homomorphisms on TROs.