Connes embedding problem

rdf:langString Connes embedding problem

rdf:langString Connes' embedding problem, formulated by Alain Connes in the 1970s, is a major problem in von Neumann algebra theory. During that time, the problem was reformulated in several different areas of mathematics. Dan Voiculescu developing his free entropy theory found that Connes’ embedding problem is related to the existence of microstates. Some results of von Neumann algebra theory can be obtained assuming positive solution to the problem. The problem is connected to some basic questions in quantum theory, which led to the realization that it also has important implications in computer science. The problem admits a number of equivalent formulations. Notably, it is equivalent to the following long standing problems: * Kirchberg's QWEP conjecture in C*-algebra theory * Tsirelson's problem in quantum information theory * The predual of any (separable) von Neumann algebra is finitely representable in the trace class. In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen announced a result in quantum complexity theory that implies a negative answer to Connes' embedding problem. However, an error was discovered in September 2020 in a result used to derive the formula; a new proof was published as a preprint in September, and a proof published in Communications of the ACM in November 2021 is currently unreviewed.