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Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors. (English) Zbl 1267.46071
Summary: We show that all the free Araki-Woods factors \(\Gamma (H_{\mathbf {R}},U_t)^{\prime\prime}\) have the complete metric approximation property. Using Ozawa-Popa’s techniques [N. Ozawa and S. Popa, Ann. Math. (2) 172, No. 1, 713–749 (2010; Zbl 1201.46054); Am. J. Math. 132, No. 3, 841–866 (2010; Zbl 1213.46053)], we then prove that every nonamenable subfactor \(\mathcal N \subset \Gamma (H_{\mathbf {R}},U_t)^{\prime\prime}\) which is the range of a normal conditional expectation has no Cartan subalgebra. We finally deduce that the type III\(_1\) factors constructed by Connes in the 1970s [A. Connes, J. Funct. Anal. 16, 415–445 (1974; Zbl 0302.46050)] can never be isomorphic to any free Araki-Woods factor, which answers a question of D. Shlyakhtenko [J. Funct. Anal. 166, No. 1, 1–47 (1999; Zbl 0951.46035)] and S. Vaes [Astérisque 299, 329–350 (2005; Zbl 1091.46037)].

MSC:
46L07 Operator spaces and completely bounded maps
46L10 General theory of von Neumann algebras
46L54 Free probability and free operator algebras
46L36 Classification of factors
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