Order Preservation in Limit Algebras

TL;DR

This paper studies operator algebras formed as direct limits of finite-dimensional algebras with order preserving embeddings, providing a classification for certain cases and exploring their structural properties.

Contribution

It offers a complete classification of direct limits of full triangular matrix algebras with order preserving embeddings and investigates their structural characterization.

Findings

Complete classification of direct limits of full triangular matrix algebras with order preserving embeddings

Analysis of operator algebras as direct limits with order preserving properties

Characterization of algebras with order preserving embeddings

Abstract

The matrix units of a digraph algebra, A, induce a relation, known as the diagonal order, on the projections in a masa in the algebra. Normalizing partial isometries in A act on these projections by conjugation; they are said to be order preserving when they respect the diagonal order. Order preserving embeddings, in turn, are those embeddings which carry order preserving normalizers to order preserving normalizers. This paper studies operator algebras which are direct limits of finite dimensional algebras with order preserving embeddings. We give a complete classification of direct limits of full triangular matrix algebras with order preserving embeddings. We also investigate the problem of characterizing algebras with order preserving embeddings.