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Amenable absorption in amalgamated free product von Neumann algebras. (English) Zbl 1406.46045
Summary: We investigate the position of amenable subalgebras in arbitrary amalgamated free product von Neumann algebras \(M=M_1\ast_{B}M_{2}\). Our main result states that, under natural analytic assumptions, any amenable subalgebra of \(M\) that has a large intersection with \(M_{1}\) is actually contained in \(M_{1}\). The proof does not rely on Popa’s asymptotic orthogonality property but on the study of nonnormal conditional expectations.

46L10 General theory of von Neumann algebras
46L09 Free products of \(C^*\)-algebras
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[1] A. Alvarez, Théorème de Kurosh pour les relations d’équivalence boréliennes, Ann. Inst. Fourier 60 (2010), 1161–1200. · Zbl 1274.37004
[2] R. Boutonnet and A. Carderi, Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups, Geom. Funct. Anal. 25 (2015), 1688–1705. · Zbl 1342.46055
[3] K. R. Davidson, \(\text{C}^{*}\)-Algebras by Example, Fields Inst. Monogr. 6, Amer. Math. Soc., Providence, 1996.
[4] D. Gaboriau, Coût des relations d’équivalence et des groupes, Invent. Math. 139 (2000), 41–98. · Zbl 0939.28012
[5] U. Haagerup, Operator-valued weights in von Neumann algebras, I, J. Funct. Anal. 32 (1979), 175–206. · Zbl 0426.46046
[6] U. Haagerup, Operator-valued weights in von Neumann algebras, II, J. Funct. Anal. 33 (1979), 339–361. · Zbl 0426.46047
[7] C. Houdayer and Y. Isono, Unique prime factorization and bicentralizer problem for a class of type III factors, Adv. Math. 305 (2017), 402–455. · Zbl 1371.46050
[8] C. Houdayer and Y. Ueda, Asymptotic structure of free product von Neumann algebras, Math. Proc. Cambridge Philos. Soc. 161 (2016), 489–516. · Zbl 1379.46047
[9] B. A. Leary, On maximal amenable subalgebras of amalgamated free product von Neumann algebras, Ph.D. dissertation, University of California–Los Angeles, Los Angeles, California, USA, 2015.
[10] N. Ozawa, A remark on amenable von Neumann subalgebras in a tracial free product, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 104. · Zbl 1351.46059
[11] N. Ozawa and S. Popa, On a class of \(\text{II}_{1}\) factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (2010), 713–749. · Zbl 1201.46054
[12] N. Ozawa and S. Popa, On a class of \(\text{II}_{1}\) factors with at most one Cartan subalgebra, II, Amer. J. Math. 132 (2010), 841–866. · Zbl 1213.46053
[13] S. Popa, Maximal injective subalgebras in factors associated with free groups, Adv. Math. 50 (1983), 27–48. · Zbl 0545.46041
[14] S. Popa, On a class of type \(\text{II_{1}}\) factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), 809–899. · Zbl 1120.46045
[15] S. Popa, Strong rigidity of \(\text{II_{1}}\) factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006), 369–408. · Zbl 1120.46043
[16] M. Takesaki, Theory of Operator Algebras, II, Encyclopaedia Math. Sci. 125, Springer, Berlin, 2003. · Zbl 1059.46031
[17] Y. Ueda, Amalgamated free products over Cartan subalgebra, Pacific J. Math. 191 (1999), 359–392. · Zbl 1030.46085
[18] Y. Ueda, Remarks on HNN extensions in operator algebras, Illinois J. Math. 52 (2008), 705–725. · Zbl 1183.46057
[19] Y. Ueda, Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math. 228 (2011), 2647–2671. · Zbl 1252.46059
[20] D.-V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, CRM Monogr. Ser. 1, Amer. Math. Soc., Providence, 1992.
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