Inclusions of ternary rings of operators and conditional expectations

TL;DR

This paper demonstrates that certain contractive idempotent maps on ternary rings of operators can be uniquely extended to conditional expectations, with explicit descriptions, including for W*-TROs, and discusses applications.

Contribution

It provides a new explicit characterization of extensions of contractive idempotent maps to conditional expectations in TROs and W*-TROs, advancing the understanding of their structure.

Findings

Unique extension of contractive idempotent maps to conditional expectations

Explicit description of the extension process

Applications to W*-TROs and related structures

Abstract

It is shown that if T is a ternary ring of operators (TRO), X is a nondegenerate sub-TRO of T and there exists a contractive idempotent surjective map P:T-->X, then P has a unique, explicitly described extension to a conditional expectation between the associated linking algebras. A version of the result for W*-TROs is also presented and some applications mentioned.