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Factoriality, type classification and fullness for free product von Neumann algebras. (English) Zbl 1252.46059
Summary: We give a complete answer to the questions of factoriality, type classification and fullness for arbitrary free products von Neumann algebras.

##### MSC:
 46L10 General theory of von Neumann algebras 46L09 Free products of $$C^*$$-algebras 46L36 Classification of factors
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##### References:
 [1] Barnett, L., Free product von Neumann algebras of type III, Proc. amer. math. soc., 123, 543-553, (1995) · Zbl 0808.46088 [2] Chifan, I.; Houdayer, C., Bass-Serre rigidity results in von Neumann algebras, Duke math. J., 153, 23-54, (2010) · Zbl 1201.46057 [3] Connes, A., Almost periodic states and factors of type III_{1}, J. funct. anal., 16, 415-445, (1974) · Zbl 0302.46050 [4] Connes, A.; Størmer, E., Homogeneity of the state space of factors of type III_{1}, J. funct. anal., 28, 187-196, (1978) · Zbl 0408.46048 [5] Dykema, K., Free products of hyperfinite von Neumann algebras and free dimension, Duke math. J., 69, 97-119, (1993) · Zbl 0784.46044 [6] Dykema, K., Factoriality and connesʼ invariant $$T(\mathcal{M})$$ for free products of von Neumann algebras, J. reine angew. math., 450, 159-180, (1994) · Zbl 0791.46037 [7] Dykema, K., Amalgamated free products of multi-matrix algebras and a construction of subfactors of a free group factor, Amer. J. math., 117, 1555-1602, (1995) · Zbl 0854.46051 [8] Dykema, K., Free products of finite-dimensional and other von Neumann algebras with respect to non-tracial states, (), 41-88 · Zbl 0871.46030 [9] Haagerup, U.; Størmer, E., Equivalence of normal states on von Neumann algebras and the flow of weights, Adv. math., 83, 180-262, (1990) · Zbl 0717.46054 [10] Houdayer, C., On some free products of von Neumann algebras which are free Araki-Woods factors, Int. math. res. not. IMRN, 23, (2007), Art. ID rnm098, 21 pp · Zbl 1133.46031 [11] Houdayer, C.; Ricard, E., Approximation properties and absence of Cartan subalgebra for free Araki-Woods factors, Adv. math., 228, 2, 764-802, (2011) · Zbl 1267.46071 [12] Jung, K., The free entropy dimension of hyperfinite von Neumann algebras, Trans. amer. math. soc., 355, 5053-5089, (2003) · Zbl 1028.46096 [13] Ocneanu, A., Actions of discrete amenable groups on von Neumann algebras, Lecture notes in math., vol. 1138, (1985), Springer-Verlag · Zbl 0608.46035 [14] Ozawa, N., Examples of groups which are not weakly amenable, (2010), preprint [15] Ozawa, N.; Popa, S., On a class of II_{1} factors with at most one Cartan subalgebra, Ann. of math. (2), 172, 713-749, (2010) · Zbl 1201.46054 [16] Pathak, P.K.; Shapiro, H.S., A characterization of certain weak^{⁎}-closed subalgebras of L∞, J. math. anal. appl., 58, 174-177, (1977) · Zbl 0347.46060 [17] Popa, S., On a problem of R.V. kadison on maximal abelian ⁎-subalgebras in factors, Invent. math., 65, 269-281, (1981/1982) · Zbl 0481.46028 [18] Popa, S., Orthogonal pairs of ⁎-subalgebras in finite von Neumann algebras, J. operator theory, 9, 253-268, (1983) · Zbl 0521.46048 [19] Popa, S., Maximal injective subalgebras in factors associated with free groups, Adv. math., 50, 27-48, (1983) · Zbl 0545.46041 [20] Shlyakhtenko, D., Free quasi-free states, Pacific J. math., 177, 329-368, (1997) · Zbl 0882.46026 [21] Shlyakhtenko, D., On the classification of full factors of type III, Trans. amer. math. soc., 356, 4143-4159, (2004) · Zbl 1050.46046 [22] Takesaki, M., Theory of operator algebras, I, Oper. alg. non-commut. geom., vol. 5, (2002), Springer Berlin, Encyclopaedia Math. Sci., vol. 124 · Zbl 0990.46034 [23] Takesaki, M., Theory of operator algebras, II, Oper. alg. non-commut. geom., vol. 6, (2003), Springer Berlin, Encyclopaedia Math. Sci., vol. 125 · Zbl 1059.46031 [24] Ueda, Y., Remarks on free products with respect to non-tracial states, Math. scand., 88, 111-125, (2001) · Zbl 1026.46048 [25] Ueda, Y., Fullness, connesʼ χ-groups, and ultra-products of amalgamated free products over Cartan subalgebras, Trans. amer. math. soc., 355, 349-371, (2003) · Zbl 1028.46097 [26] Ueda, Y., HNN extensions of von Neumann algebras, J. funct. anal., 225, 383-426, (2005) · Zbl 1088.46034 [27] Y. Ueda, On type III_{1} factors arising as free products, Math. Res. Lett., in press. · Zbl 1243.46053 [28] Voiculescu, D., The analogues of entropy and of fisherʼs information measure in free probability theory. III. the absence of Cartan subalgebras, Geom. funct. anal., 6, 172-199, (1996) · Zbl 0856.60012 [29] Voiculescu, D.-V.; Dykema, K.-J.; Nica, A., Free random variables, CRM monogr. ser., vol. 1, (1992), Amer. Math. Soc. Providence, RI · Zbl 0795.46049 [30] Woeden, B.J., Normalcy in von Neumann algebras, Proc. London math. soc. (3), 27, 88-100, (1973) · Zbl 0264.46065
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