On the Toeplitz-Jacobson algebra and direct finiteness

TL;DR

This paper explores the structure and representation theory of the Toeplitz algebra, providing new theorems, classifying ideals, and discussing its relevance to major conjectures like Kaplansky's and Connes' embedding conjecture.

Contribution

It offers new structural and homological results for the Toeplitz algebra and connects its module theory to important open problems in mathematics.

Findings

Complete classification of one-sided ideals

New structure and homological theorems

Connections to Kaplansky's and Connes' conjectures

Abstract

We study the representation theory of the algebraic Toeplitz algebra $R={\mathbb K}\langle x,y\rangle/\langle xy-1\rangle$, give a few new structure and homological theorems, completely determine one-sided ideals and survey and re-obtain results from the existing literature as well. We discuss its connection to Kaplansky's direct finiteness conjecture, and a possible approach to it based on the module theory of $R$. In addition, we discuss the conjecture's connections to several central problems in mathematics, including Connes' embedding conjecture.