## Kyoto Journal of Mathematics

### Amenable absorption in amalgamated free product von Neumann algebras

#### Abstract

We investigate the position of amenable subalgebras in arbitrary amalga- mated 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.

#### Article information

Source
Kyoto J. Math., Volume 58, Number 3 (2018), 583-593.

Dates
Revised: 16 November 2016
Accepted: 21 November 2016
First available in Project Euclid: 5 June 2018

https://projecteuclid.org/euclid.kjm/1528185692

Digital Object Identifier
doi:10.1215/21562261-2017-0030

Mathematical Reviews number (MathSciNet)
MR3843391

Zentralblatt MATH identifier
06959092

#### Citation

Boutonnet, Rémi; Houdayer, Cyril. Amenable absorption in amalgamated free product von Neumann algebras. Kyoto J. Math. 58 (2018), no. 3, 583--593. doi:10.1215/21562261-2017-0030. https://projecteuclid.org/euclid.kjm/1528185692

#### References

• [1] A. Alvarez, Théorème de Kurosh pour les relations d’équivalence boréliennes, Ann. Inst. Fourier 60 (2010), 1161–1200.
• [2] R. Boutonnet and A. Carderi, Maximal amenable von Neumann subalgebras arising from maximal amenable subgroups, Geom. Funct. Anal. 25 (2015), 1688–1705.
• [3] K. R. Davidson, $\mathrm{C}^{*}$-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.
• [5] U. Haagerup, Operator-valued weights in von Neumann algebras, I, J. Funct. Anal. 32 (1979), 175–206.
• [6] U. Haagerup, Operator-valued weights in von Neumann algebras, II, J. Funct. Anal. 33 (1979), 339–361.
• [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.
• [8] C. Houdayer and Y. Ueda, Asymptotic structure of free product von Neumann algebras, Math. Proc. Cambridge Philos. Soc. 161 (2016), 489–516.
• [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.
• [11] N. Ozawa and S. Popa, On a class of $\mathrm{II}_{1}$ factors with at most one Cartan subalgebra, Ann. of Math. (2) 172 (2010), 713–749.
• [12] N. Ozawa and S. Popa, On a class of $\mathrm{II}_{1}$ factors with at most one Cartan subalgebra, II, Amer. J. Math. 132 (2010), 841–866.
• [13] S. Popa, Maximal injective subalgebras in factors associated with free groups, Adv. Math. 50 (1983), 27–48.
• [14] S. Popa, On a class of type $\mathrm{II_{1}}$ factors with Betti numbers invariants, Ann. of Math. (2) 163 (2006), 809–899.
• [15] S. Popa, Strong rigidity of $\mathrm{II_{1}}$ factors arising from malleable actions of w-rigid groups, I, Invent. Math. 165 (2006), 369–408.
• [16] M. Takesaki, Theory of Operator Algebras, II, Encyclopaedia Math. Sci. 125, Springer, Berlin, 2003.
• [17] Y. Ueda, Amalgamated free products over Cartan subalgebra, Pacific J. Math. 191 (1999), 359–392.
• [18] Y. Ueda, Remarks on HNN extensions in operator algebras, Illinois J. Math. 52 (2008), 705–725.
• [19] Y. Ueda, Factoriality, type classification and fullness for free product von Neumann algebras, Adv. Math. 228 (2011), 2647–2671.
• [20] D.-V. Voiculescu, K. J. Dykema, and A. Nica, Free Random Variables, CRM Monogr. Ser. 1, Amer. Math. Soc., Providence, 1992.