On the Toeplitz-Jacobson algebra and direct finiteness

Abstract

We study the representation theory of the algebraic Toeplitz algebra $R=\mathbb{K}⟨x,y⟩/⟨xy-1⟩$, 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.