标题： 关于Toeplitz-Jacobson代数和直诣有限性 显示英文标题

标题： On the Toeplitz-Jacobson algebra and direct finiteness

摘要： 我们研究了代数Toeplitz代数$R={\mathbb K}\langle x,y\rangle/\langle xy-1\rangle$的表示理论，给出了若干新的结构和同调定理，完全确定了一侧理想，并回顾和重新获得了现有文献中的结果。 我们讨论了它与Kaplansky的直接有限性猜想之间的联系，并基于$R$的模理论提出了一种可能的方法。 此外，我们讨论了该猜想与数学中几个中心问题的联系，包括Connes的嵌入猜想。 显示英文摘要

摘要： 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.