Universal properties of \(L(\mathbb{F}_ \infty)\) in subfactor theory.
type: Item
instance of: scholarly article
title: Universal properties of \(L(\mathbb{F}_ \infty)\) in subfactor theory.
zbMATH Open document ID: 1079.46043
DOI: 10.1007/BF02392965
published in: Acta Mathematica
publication date: 2004-11-15T00:00:00Z
review text: 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 \(\wi…
reviewed by: Florin P. Boca
Mathematics Subject Classification ID: 46L37
Mathematics Subject Classification ID: 46L09
zbMATH DE Number: 2115788
zbMATH Keywords: subfactors
zbMATH Keywords: standard lattice
zbMATH Keywords: free group factors
zbMATH Keywords: commuting squares
zbMATH Keywords: amalgamated free products
author: Sorin Popa
author: D. Shlyakhtenko
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2115788