Convex sets associated with von Neumann algebras and Connes' approximate embedding problem
Connes' approximate embedding problem asks whether any countably generated type \(\text{II}_1\) factor \(M\) with trace \(\tau\) can be approximately embedded in the hyperfinite type \(\text{II}_1\) factor. Solving this problem in the affirmative amounts to showing that given any integers \(N\) and \(p\), any elements \(x_1,\dots, x_N\) in \(M\) and any \(\varepsilon> 0\), one can find \(k\) and matrices \(X_1,\dots, X_N\) in the matrix algebra \(M_k(\mathbb{C})\) such that \(|\tau(x_{i_1},\dots, x_{i_s})- \text{tr}(X_{i_1},\dots, X_{i_s})|< \varepsilon\) for every \(i_1,\dots, i_p\) in \(\{1,\dots, N\}\) and for every \(s\) with \(1\leq s\leq p\), where tr is the normalized trace on \(M_k(\mathbb{C})\). In this paper, the author shows that this is always possible for \(s= 2,3\). An affirmative solution of Connes' problem would follow if one could also prove the above for \(s= 4\).
scientific article; zbMATH DE number 1489494