Open projections in operator algebras I: Comparison theory

TL;DR

This paper extends the theory of comparison and equivalence in C*-algebras to more general operator algebras using open projections and module isomorphisms, introducing new inner ideals and a matching theory of open partial isometries.

Contribution

It introduces a new framework for comparison theory in operator algebras, generalizing concepts of open projections and partial isometries, and defines novel inner ideals.

Findings

Characterization of a new class of inner ideals in operator algebras

Development of a matching theory for open partial isometries

Generalization of comparison and equivalence concepts in broader algebraic settings

Abstract

We begin a program of generalizing basic elements of the theory of comparison, equivalence, and subequivalence, of elements in C*-algebras, to the setting of more general algebras. In particular, we follow the recent lead of Lin, Ortega, Rordam, and Thiel of studying these equivalences, etc., in terms of open projections or module isomorphisms. We also define and characterize a new class of inner ideals in operator algebras, and develop a matching theory of open partial isometries in operator ideals which simultaneously generalize the open projections in operator algebras (in the sense of the authors and Hay), and the open partial isometries (tripotents) introduced by the authors.