Ultraproducts, QWEP von Neumann algebras, and the Effros-Maréchal topology (Q294269)

Classical work of \textit{E. Kirchberg} [Invent. Math. 112, No. 3, 449--489 (1993; Zbl 0803.46071)] established that a positive answer to the celebrated Connes problem, concerning the embeddability of every II\(_1\) factor with separable predual into an ultrapower \(R^\omega\) of the injective II\(_1\) factor \(R\), is equivalent to every \(C^*\)-algebra having the QWEP (quotient weak expectation property), or with the equality between the minimal and maximal \(C^*\)-norms on the tensor product \(C^*({\mathbb F}_\infty) \otimes C^* ({\mathbb F}_\infty)\). Subsequent work of \textit{U. Haagerup} and \textit{C. Winsløw} [J. Funct. Anal. 171, No. 2, 401--431 (2000; Zbl 0982.46045)] produced an equivalent elegant formulation, showing that a II\(_1\) factor \(N\), acting on a separable infinite-dimensional Hilbert space \(H\), is \(R^\omega\)-embeddable if and only if \(N\) belongs to \(\overline{\mathcal F}_{inj}\), the closure of the set of injective factors in the Effros-Maréchal topology on the set \(\text{vN}(H)\) of all von Neumann algebras acting on \(H\).NEWLINENEWLINEThe paper under review provides valuable new insight into this circle of problems, offering reformulations of the QWEP for von Neumann algebras in terms of the Effros-Maréchal topology, of embeddability into ultraproducts of type III hyperfinite factors, or of the standard form of a von Neumann algebra. More precisely, it is shown that the following conditions are equivalent for some \(M\in \text{vN} (H)\): {\parindent=6mm \begin{itemize}\item[(1)] \(M\) has the QWEP. \item[(2)] \(M \in \overline{\mathcal F}_{inj}\). \item[(3)] \(M\) embeds into an ultraproduct \({\mathcal R}^\omega\) with a normal faithful conditional expectation from \({\mathcal R}^\omega\) onto the image of \(M\), where \({\mathcal R}\) is one of the injective type III\(_\lambda\) factors, \(0<\lambda \leq 1\). \item[(4)] For every \(\varepsilon >0\), \(n\geq 1\) and vectors \(\xi_1,\dots,\xi_n \in {\mathcal P}_M^\natural\), there exist \(a_1,\dots,a_n \in M_k({\mathbb C})_+\) for some \(k\geq 1\) such that NEWLINE\[NEWLINE| \langle \xi_i,\xi_j \rangle - \text{tr}_k (a_i a_j) | < \varepsilon ,\quad i,j=1,\dots,n ,NEWLINE\]NEWLINE where \({\mathcal P}_M^\natural\) denotes the natural cone in the standard form of \(M\) and \(\text{tr}_k\) is the tracial state on \(M_k ({\mathbb C})\). NEWLINENEWLINE\end{itemize}} The proofs are intricate but the paper is very well written overall.