Structure of bicentralizer algebras and inclusions of type III factors

Cyril Houdayer
Université Paris-Sud (Orsay)

Connes' bicentralizer problem (CBP) asks whether every type ${\rm III_{1}}$ factor has a trivial bicentralizer. Haagerup solved CBP for amenable type ${\rm III_{1}}$ 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 ${\rm III_{1}}$ 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 ${\rm III_{1}}$ factors $N \subset M$. We construct a canonical flow $\beta : \R \curvearrowright {\mathrm B}(N \subset M)$ 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.

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