In his celebrated paper in 1976, A. Connes casually remarked that any finite von Neumann algebra ought to be embedded into an ultraproduct of matrix algebras, which is now known as the Connes embedding conjecture or problem. This conjecture became one of the central open problems in the field of operator algebras since E. Kirchberg’s seminal work in 1993 that proves it is equivalent to a variety of other seemingly totally unrelated but important conjectures in the field. Since then, many more equivalents of the conjecture have been found, also in some other branches of mathematics such as noncommutative real algebraic geometry and quantum information theory. In this note, we present a survey of this conjecture with a focus on the algebraic aspects of it.

中文翻译：

---

关于Connes嵌入猜想

康纳斯（A. ​​Connes）在1976年发表的著名论文中随意地指出，任何有限的冯·诺依曼代数都应嵌入矩阵代数的超积中，该矩阵现在被称为康尼斯嵌入猜想或问题。自从1993年E.基希贝格（E.Kirchberg）的开创性工作证明这一假设等同于该领域中其他许多似乎毫无关联但重要的猜想以来，这一猜想就成为算子代数领域的主要开放问题之一。从那时起，在数学的其他一些分支中，例如非交换实数代数几何和量子信息理论，也找到了猜想的更多等价物。在本说明中，我们将对此猜想进行一个调查，重点是它的代数方面。