On the Toeplitz-Jacobson algebra and direct finiteness
Abstract
We study the representation theory of the algebraic Toeplitz algebra R= 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.
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