Contractive projections and real positive maps on operator algebras

TL;DR

This paper investigates contractive projections, isometries, and real positive maps on operator algebras, extending classical results and establishing new foundational theorems, including a Banach-Stone type theorem for operator algebra isometries.

Contribution

It generalizes classical results on contractive projections to Jordan operator algebras and introduces new results on real positive maps and a Banach-Stone type theorem.

Findings

Generalized classical results on contractive projections to Jordan operator algebras

Proved new foundational results on real positive maps

Established a Banach-Stone type theorem for operator algebra isometries

Abstract

We study contractive projections, isometries, and real positive maps on algebras of operators on a Hilbert space. For example we find generalizations and variants of certain classical results on contractive projections on C*-algebras and JB-algebras due to Choi, Effros, St{\o}rmer, Friedman and Russo, and others. In fact most of our arguments generalize to contractive `real positive' projections on Jordan operator algebras, that is on a norm-closed space A of operators on a Hilbert space which are closedunder the Jordan product. We also prove many new general results on real positive maps which are foundational to the study of such maps, and of interest in their own right. We also prove a new Banach-Stone type theorem for isometries between operator algebras or Jordan operator algebras. An application of this is given to the characterization of symmetric real positive projections.