Universal properties of \(L(\mathbb{F}_ \infty)\) in subfactor theory.
Higher relative commutants \(M_i^\prime \cap M_j\) are finite-dimensional \(C^*\)-algebras obtained from the Jones tower \(N=M_{-1} \subset M=M_0 \subset M_1 \subset \ldots\) of an inclusion \(N\subset M\) of von Neumann factors of type II\(_1\). The lattice \(\{ M_i^\prime \cap M_j\}_{-1\leq i\leq j}\) plays a central role in the classification of subfactors. An axiomatic characterization of these objects was given by the first author [\textit{S. Popa}, Invent. Math. 120, No. 3, 427--445 (1995; Zbl 0831.46069)]. Such an abstract object is called a standard lattice. The construction of a II\(_1\) factor inclusion with prescribed lattice of higher relative commutants uses amalgamated free products and depends on the choice of a von Neumann algebra \(Q\). The paper under review contains a number of important contributions to the study of standard lattices and subfactors in free group factors. The type II\(_1\) group factor \(L({\mathbb F}_\infty)\) of the free group on a countable infinite number of generators is shown to have subfactor universal properties, in the sense that its subfactors realize all possible standard lattices in a functorial way. More precisely, the following results are proved: Theorem 1. Any standard lattice can be realized as the system of higher relative commutants of a type II\(_1\) subfactor \(N\subset M\), where both \(N\) and \(M\) are isomorphic to the free group factor \(L({\mathbb F}_\infty)\). Moreover, the construction of subfactors \(N\subset M\) can be chosen to be a functor from the category of standard lattices (with commuting square inclusions as morphisms) to the category of subfactors (with commuting square inclusions as morphisms). Theorem 2. The type II\(_1\) factors appearing in the inclusions previously constructed by Popa for the initial data \(Q=L({\mathbb F}_\infty)\) are all isomorphic to \(L({\mathbb F}_\infty)\). Theorem 3. Given an arbitrary inclusion of II\(_1\) factors \(N\subset M\), there exists an inclusion \(\widehat{N}\subset \hat{M}\) with the same standard lattice as \(N\subset M\) and so that \(\widehat{M}\cong \hat{N}\cong M\ast L({\mathbb F}_\infty)\). These results generalize and put in a natural abstract context previous work from \textit{S. Popa} [Invent. Math. 111, No. 2, 375--405 (1993; Zbl 0787.46047), Invent. Math. 120, No. 3, 427--445 (1995; Zbl 0831.46069)], \textit{F. Radulescu} [Invent. Math. 115, No. 2, 347--389 (1994; Zbl 0861.46038)], \textit{K. Dykema} [Am. Math. J. 117, No. 6, 1555--1602 (1995; Zbl 0854.46051)], \textit{D. Shlyakhtenko} [J. Funct. Anal. 166, No. 1, 1--47 (1999; Zbl 0951.46035)], and \textit{D. Shlyakhtenko} and \textit{Y. Ueda} [J. Reine Angew. Math. 548, 149--166 (2002; Zbl 1009.46035)].
scientific article; zbMATH DE number 2115788