Abstract
Structure of bicentralizer algebras and inclusions of type III factors
Cyril Houdayer
Université Paris-Sud (Orsay)
Connes' bicentralizer problem (CBP) asks whether every type \( \mathrm{III}_1 \) factor has a trivial bicentralizer. Haagerup solved CBP for amenable type \( \mathrm{III}_1 \) factors, thus completing Connes' classification of amenable factors. CBP is known to have a positive solution in some particular cases but remains wide open for arbitrary nonamenable type \( \mathrm{III}_1 \) factors.
Motivated by CBP, we investigate the structure of the (relative) bicentralizer algebra \( \mathrm{B}(N \subset M) \) associated with an irreducible inclusion of type \( \mathrm{III}_1 \) factors \( N \subset M \). We construct a canonical flow \( \beta : \mathbb{R} \curvearrowright \mathrm{B}(N \subset M) \) that does not depend on the choice of states and relate the ergodicity of the flow \( \beta \) to the existence of amenable subfactors \( P \subset N \) that are irreducible in \( M \).
This also provides new results on the structure of the bicentralizer algebra \( \mathrm{B}(M) \) in the case \( N = M \). When the inclusion \( N \subset M \) is discrete, we prove a relative version of Haagerup's bicentralizer theorem and use it to solve Kadison's problem when \( N \) is amenable.
This is joint work with Hiroshi Ando, Uffe Haagerup and Amine Marrakchi.
Motivated by CBP, we investigate the structure of the (relative) bicentralizer algebra \( \mathrm{B}(N \subset M) \) associated with an irreducible inclusion of type \( \mathrm{III}_1 \) factors \( N \subset M \). We construct a canonical flow \( \beta : \mathbb{R} \curvearrowright \mathrm{B}(N \subset M) \) that does not depend on the choice of states and relate the ergodicity of the flow \( \beta \) to the existence of amenable subfactors \( P \subset N \) that are irreducible in \( M \).
This also provides new results on the structure of the bicentralizer algebra \( \mathrm{B}(M) \) in the case \( N = M \). When the inclusion \( N \subset M \) is discrete, we prove a relative version of Haagerup's bicentralizer theorem and use it to solve Kadison's problem when \( N \) is amenable.
This is joint work with Hiroshi Ando, Uffe Haagerup and Amine Marrakchi.