×

zbMATH — the first resource for mathematics

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.

MSC:
46L10 General theory of von Neumann algebras
46L09 Free products of \(C^*\)-algebras
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[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.
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.