Mathematics – Operator Algebras
Scientific paper
2006-06-12
Int. Math. Res. Notices. (2007) Vol. 2007, article ID rnm098, 21 pages
Mathematics
Operator Algebras
14 pages
Scientific paper
We prove that certain free products of factors of type ${\rm I}$ and other von Neumann algebras with respect to nontracial, almost periodic states are almost periodic free Araki-Woods factors. In particular, they have the free absorption property and Connes' Sd invariant completely classifies these free products. For example, for $\lambda, \mu \in ]0, 1[$, we show that $$(M_2(\C), \omega_{\lambda}) \ast (M_2(\C), \omega_{\mu})$$ is isomorphic to the free Araki-Woods factor whose Sd invariant is the subgroup of $\R^*_+$ generated by $\lambda$ and $\mu$. Our proofs are based on algebraic techniques and amalgamated free products. These results give some answers to questions of Dykema and Shlyakhtenko.
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