zbMATH — the first resource for mathematics

Gamma stability in free product von Neumann algebras. (English) Zbl 1328.46046
Summary: Let \((M,\varphi)=(M_1,\varphi_1)\ast(M_2,\varphi_2)\) be a free product of arbitrary von Neumann algebras endowed with faithful normal states. Assume that the centralizer \(M_1^{\varphi_1}\) is diffuse. We first show that any intermediate subalgebra \(M_1\subset Q\subset M\) which has nontrivial central sequences in \(M\) is necessarily equal to \(M_1\). Then we obtain a general structural result for all the intermediate subalgebras \(M_1\subset Q\subset M\) with expectation. We deduce that any diffuse amenable von Neumann algebra can be concretely realized as a maximal amenable subalgebra with expectation inside a full nonamenable type \(\mathrm{III}_1\) factor. This provides the first class of concrete maximal amenable subalgebras in the framework of type III factors. We finally strengthen all these results in the case of tracial free product von Neumann algebras.

46L10 General theory of von Neumann algebras
46L09 Free products of \(C^*\)-algebras
46L36 Classification of factors
Full Text: DOI arXiv
[1] Ando, H.; Haagerup, U., Ultraproducts of von Neumann algebras, J. Funct. Anal., 266, 6842-6913, (2014) · Zbl 1305.46049
[2] Barnett, L., Free product von Neumann algebras of type III, Proc. Amer. Math. Soc., 123, 543-553, (1995) · Zbl 0808.46088
[3] Boutonnet, R., Carderi, A.: Maximal amenable subalgebras of von Neumann algebras associated with hyperbolic groups. arXiv:1310.5864 · Zbl 1369.46052
[4] Brothier, A.: The cup subalgebra of a II_{1} factor given by a subfactor planar algebra is maximal amenable. Pacific J. Math. (to appear). arXiv:1210.8091 · Zbl 1320.46046
[5] Cameron, J.; Fang, J.; Ravichandran, M.; White, S., The radial masa in a free group factor is maximal injective, J. Lond. Math. Soc., 82, 787-809, (2010) · Zbl 1237.46043
[6] Connes, A., Une classification des facteurs de type III, Ann. Sci. École Norm. Sup., 6, 133-252, (1973) · Zbl 0274.46050
[7] Connes, A., Almost periodic states and factors of type III_{1}, J. Funct. Anal., 16, 415-445, (1974) · Zbl 0302.46050
[8] Connes, A., Outer conjugacy classes of automorphisms of factors, Ann. Sci. École Norm. Sup., 8, 383-419, (1975) · Zbl 0342.46052
[9] Connes, A., Classification of injective factors, Ann. Math., 104, 73-115, (1976) · Zbl 0343.46042
[10] Connes, A., Factors of type III_{1}, property \({{{\rm L}'_λ}}\) and closure of inner automorphisms, J. Operator Theory, 14, 189-211, (1985) · Zbl 0597.46063
[11] Dykema, K., Factoriality and connes’ invariant \({T({\mathcal{M}})}\) for free products of von Neumann algebras, J. Reine Angew. Math., 450, 159-180, (1994) · Zbl 0791.46037
[12] Fang, J., On maximal injective subalgebras of tensor products of von Neumann algebras, J. Funct. Anal., 244, 277-288, (2007) · Zbl 1121.46046
[13] Gao, M., On maximal injective subalgebras, Proc. Am. Math. Soc., 138, 2065-2070, (2010) · Zbl 1196.46046
[14] Ge, L., On maximal injective subalgebras of factors, Adv. Math., 118, 34-70, (1996) · Zbl 0866.46039
[15] Haagerup, U., Connes’ bicentralizer problem and uniqueness of the injective factor of type III_{1}, Acta Math., 158, 95-148, (1987) · Zbl 0628.46061
[16] Houdayer, C., Construction of type II_{1} factors with prescribed countable fundamental group, J. Reine Angew. Math., 634, 169-207, (2009) · Zbl 1209.46038
[17] Houdayer, C., A class of II_{1} factors with an exotic abelian maximal amenable subalgebra, Trans. Am. Math. Soc., 366, 3693-3707, (2014) · Zbl 1303.46044
[18] Houdayer, C., Structure of II_{1} factors arising from free bogoljubov actions of arbitrary groups, Adv. Math., 260, 414-457, (2014) · Zbl 1297.46042
[19] Haagerup, U.; Stø rmer, E., Equivalence of normal states on von Neumann algebras and the flow of weights, Adv. Math., 83, 180-262, (1990) · Zbl 0717.46054
[20] Ioana, A.: Cartan subalgebras of amalgamated free product II_{1} factors. Ann. Sci. École Norm. Sup. (to appear) arXiv:1207.00541 · Zbl 0791.46037
[21] Ioana, A.; Peterson, J.; Popa, S., Amalgamated free products of \(w\)-rigid factors and calculation of their symmetry groups, Acta Math., 200, 85-153, (2008) · Zbl 1149.46047
[22] Jolissaint, P.: Maximal injective and mixing masas in group factors. arXiv:1004.0128 · Zbl 0545.46041
[23] Murray, F.; Neumann, J., Rings of operators. IV_{1}, Ann. Math., 44, 716-808, (1943) · Zbl 0060.26903
[24] Ocneanu, A.: Actions of discrete amenable groups on von Neumann algebras. In: Lecture Notes in Mathematics, vol. 1138. Springer, Berlin (1985) · Zbl 0608.46035
[25] Peterson, J., L\^{}{2}-rigidity in von Neumann algebras, Invent. Math., 175, 417-433, (2009) · Zbl 1170.46053
[26] Popa, S., Maximal injective subalgebras in factors associated with free groups, Adv. Math., 50, 27-48, (1983) · Zbl 0545.46041
[27] Popa, S., On a class of type II_{1} factors with Betti numbers invariants, Ann. Math., 163, 809-899, (2006) · Zbl 1120.46045
[28] Popa, S.: Strong rigidity of II_{1} factors arising from malleable actions of \(w\)-rigid groups I and II. Invent. Math. 165, 369-408, 409-451 (2006) · Zbl 1120.46043
[29] Popa, S., A II_{1} factor approach to the kadison-Singer problem, Comm. Math. Phys., 332, 379-414, (2014) · Zbl 1306.46060
[30] Shen, J., Maximal injective subalgebras of tensor products of free group factors, J. Funct. Anal., 240, 334-348, (2006) · Zbl 1113.46061
[31] Takesaki, M.: Theory of operator algebras. II. Encyclopaedia of Mathematical Sciences, vol. 125. Operator Algebras and Non-commutative Geometry, 6. Springer, Berlin (2003) · Zbl 1059.46032
[32] Ueda, Y., Amalgamated free products over Cartan subalgebra, Pacific J. Math., 191, 359-392, (1999) · Zbl 1030.46085
[33] Ueda, Y., Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math., 228, 2647-2671, (2011) · Zbl 1252.46059
[34] Voiculescu, D.-V.: Symmetries of some reduced free product C\^{}{*}-algebras. In: Operator Algebras and Their Connections with Topology and Ergodic Theory. Lecture Notes in Mathematics, vol. 1132, pp. 556-588. Springer, Berlin (1985) · Zbl 0302.46050
[35] Voiculescu, D.-V., Dykema, K.J., Nica, A.: Free random variables. In: CRM Monograph Series, vol. 1. American Mathematical Society, Providence (1992) · Zbl 0795.46049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.