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Classification of a family of non-almost-periodic free Araki-Woods factors. (English) Zbl 1434.46037
Summary: We obtain a complete classification of a large class of non-almost-periodic free Araki-Woods factors \(\Gamma(\mu, m)^{\prime\prime}\) up to isomorphism. We do this by showing that free Araki-Woods factors \(\Gamma(\mu, m)^{\prime\prime}\) arising from finite symmetric Borel measures \(\mu\) on \(\mathbb{R}\) whose atomic part \(\mu_a\) is nonzero and not concentrated on \(\{0\}\) have the joint measure class \(\mathcal C(\bigvee_{k \geq 1} \mu^{\ast k})\) as an invariant. Our key technical result is a deformation/rigidity criterion for the unitary conjugacy of two faithful normal states. We use this to also deduce rigidity and classification theorems for free product von Neumann algebras.

MSC:
46L36 Classification of factors
46L10 General theory of von Neumann algebras
46L09 Free products of \(C^*\)-algebras
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[1] Boutonnet, R., Houdayer, C.: Structure of modular invariant subalgebras in free Araki- Woods factors. Anal. PDE 9, 1989-1998 (2016)Zbl 1369.46053 MR 3599523 · Zbl 1369.46053
[2] Connes, A.: Une classification des facteurs de type III. Ann. Sci. ´Ecole Norm. Sup. 6, 133-252 (1973)Zbl 0274.46050 MR 0341115 Classification of a family of non-almost-periodic free Araki-Woods factors3141
[3] Connes, A.: Almost periodic states and factors of type III1. J. Funct. Anal. 16, 415-445 (1974)Zbl 0302.46050 MR 0358374 · Zbl 0302.46050
[4] Connes, A.: Classification des facteurs. In: Operator Algebras and Applications, Part 2 (Kingston, 1980), Proc. Sympos. Pure Math. 38, Amer. Math. Soc., Providence, 43-109 (1982)Zbl 0503.46043 MR 0679497
[5] Dykema, K.: Free products of hyperfinite von Neumann algebras and free dimension. Duke Math. J. 69, 97-119 (1993)Zbl 0784.46044 MR 1201693 · Zbl 0784.46044
[6] Dykema, K.: Free products of finite-dimensional and other von Neumann algebras with respect to non-tracial states. In: Free Probability Theory (Waterloo, ON, 1995), Fields Inst. Comm. 12, Amer. Math. Soc., Providence, 41-88 (1997)Zbl 0871.46030 MR 1426835 · Zbl 0871.46030
[7] Haagerup, U.: Operator-valued weights in von Neumann algebras. I. J. Funct. Anal. 32, 175-206 (1979)Zbl 0426.46046 MR 0534673 · Zbl 0426.46046
[8] Haagerup, U.: Connes’ bicentralizer problem and uniqueness of the injective factor of type III1. Acta Math. 69, 95-148 (1986)Zbl 0628.46061 MR 0880070 · Zbl 0628.46061
[9] Hayes, B.: 1-bounded entropy and regularity problems in von Neumann algebras. Int. Math. Res. Notices 2018, 57-137Zbl 07013391 MR 3801429 · Zbl 1415.46039
[10] Houdayer, C.: On some free products of von Neumann algebras which are free Araki- Woods factors. Int. Math. Res. Notices 2007, art. rnm098, 21 pp.Zbl 1133.46031 MR 2377217 · Zbl 1133.46031
[11] Houdayer, C.: Free Araki-Woods factors and Connes’ bicentralizer problem. Proc. Amer. Math. Soc. 137, 3749-3755 (2009)Zbl 1183.46060 MR 2529883 · Zbl 1183.46060
[12] Houdayer, C.: Structural results for free Araki-Woods factors and their continuous cores. J. Inst. Math. Jussieu 9, 741-767 (2010)Zbl 1207.46057 MR 2684260 · Zbl 1207.46057
[13] Houdayer, C., Isono, Y.: Unique prime factorization and bicentralizer problem for a class of type III factors. Adv. Math. 305, 402-455 (2017)Zbl 1371.46050 · Zbl 1371.46050
[14] Houdayer, C., Ricard, ´E.: Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors. Adv. Math. 228, 764-802 (2011)Zbl 1267.46071 MR 2822210 · Zbl 1267.46071
[15] Houdayer, C., Ueda, Y.: Rigidity of free product von Neumann algebras. Compos. Math. 152, 2461-2492 (2016)Zbl 1379.46046 MR 3594283 · Zbl 1379.46046
[16] Izumi, M., Longo, R., Popa, S.: A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras. J. Funct. Anal. 155, 25-63 (1998)Zbl 0915.46051 MR 1622812 · Zbl 0915.46051
[17] Krieger, W.: On ergodic flows and the isomorphism of factors. Math. Ann. 223, 19-70 (1976)Zbl 0332.46045 MR 0415341 · Zbl 0332.46045
[18] Lema´nczyk, M., Parreau, F.: On the disjointness problem for Gaussian automorphisms. Proc. Amer. Math. Soc. 127, 2073-2081 (1999)Zbl 0923.28007 MR 1486742 · Zbl 0923.28007
[19] Ozawa, N.: Solid von Neumann algebras. Acta Math. 192, 111-117 (2004) Zbl 1072.46040 MR 2079600 · Zbl 1072.46040
[20] Popa, S.: On a class of type II1factors with Betti numbers invariants. Ann. of Math. 163, 809-899 (2006)Zbl 1120.46045 MR 2215135 · Zbl 1120.46045
[21] Popa, S.: Strong rigidity of II1factors arising from malleable actions of w-rigid groups, I. Invent. Math. 165, 369-408 (2006)Zbl 1120.46043 MR 2231961 · Zbl 1120.46043
[22] Powers, R. T.: Representations of uniformly hyperfinite algebras and their associated von Neumann rings. Ann. of Math. 86, 138-171 (1967)Zbl 0157.20605 MR 0218905 · Zbl 0157.20605
[23] R˘adulescu, F.: The fundamental group of the von Neumann algebra of a free group with infinitely many generators is R+ {0}. J. Amer. Math. Soc. 5, 517-532 (1992) Zbl 0791.46036 MR 1142260 3142Cyril Houdayer et al. · Zbl 0791.46036
[24] Rudin, W.: Fourier Analysis on Groups. Interscience Tracts in Pure Appl. Math. 12, Wiley, New York (1962)Zbl 0107.09603 MR 0152834
[25] Shlyakhtenko, D.: Free quasi-free states. Pacific J. Math. 177, 329-368 (1997) Zbl 0882.46026 MR 1444786 · Zbl 0882.46026
[26] Shlyakhtenko, D.: Some applications of freeness with amalgamation. J. Reine Angew. Math. 500, 191-212 (1998)Zbl 0926.46046 MR 1637501 · Zbl 0926.46046
[27] Shlyakhtenko, D.: A-valued semicircular systems. J. Funct. Anal. 166, 1-47 (1999) Zbl 0951.46035 MR 1704661 · Zbl 0951.46035
[28] Shlyakhtenko, D.: On the classification of full factors of type III. Trans. Amer. Math. Soc. 356, 4143-4159 (2004)Zbl 1050.46046 MR 2058841 · Zbl 1050.46046
[29] Shlyakhtenko, D.: On multiplicity and free absorption for free Araki-Woods factors. arXiv:math/0302217(2003)
[30] Takesaki, M.: Theory of Operator Algebras. II. Encyclopaedia Math. Sci. 125, Operator Algebras and Non-commutative Geometry 6, Springer, Berlin (2003)Zbl 1059.46031 MR 1943006
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