On maximal injective subalgebras
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Abstract:
Let $\mathcal {A}_i$ be a type $I$ von Neumann subalgebra in a type $II_1$ factor $\mathcal {M}_i$ with the faithful trace $\tau _i$ such that $\mathcal {A}_i’\cap \mathcal {M}_i\subseteq \mathcal {A}_i$, for $i=1, 2, \cdots$. Moreover, suppose $\mathcal {A}_i$ has the asymptotically orthogonal property in $\mathcal {M}_i$ after tensoring the finite von Neumann algebra $\otimes _{j\ne i}\mathcal {M}_j$, for all $i=1,2,\cdots$. Then we show that $\otimes _{i=1}^\infty \mathcal {A}_i$ is maximal injective in the infinite tensor product von Neumann algebra $\otimes _{i=1}^\infty \mathcal M_i$. As a consequence, we get the following result. Let $\{\mathbb {F}_{n_i};i=1,2, \cdots \}$ be a sequence of free groups with $n_i$ ($>1$) generators. Let $\mathcal {A}_i$ be the masa of group von Neumann algebra $\mathcal {L}_{\mathbb {F}_{n_i}}$ generated by a generator of $\mathbb {F}_{n_i}$ or by the sum of all generators and their inverses of the group. Then $\otimes _{i=1}^\infty \mathcal {A}_i$ is maximal injective in the infinite tensor product von Neumann algebra $\otimes _{i=1}^\infty \mathcal {L}_{\mathbb {F}_{n_i}}$.References
- J. Cameron, J. Fang, M. Ravichandran, and S. White. The radical masa in a free group factor is maximal injective. arXiv: 0810.3906v1[math.OA], 21 Oct. 2008.
- A. Connes, Classification of injective factors. Cases $II_{1},$ $II_{\infty },$ $III_{\lambda },$ $\lambda \not =1$, Ann. of Math. (2) 104 (1976), no. 1, 73–115. MR 454659, DOI 10.2307/1971057
- Junsheng Fang, On maximal injective subalgebras of tensor products of von Neumann algebras, J. Funct. Anal. 244 (2007), no. 1, 277–288. MR 2294484, DOI 10.1016/j.jfa.2006.12.006
- Li Ming Ge, On “Problems on von Neumann algebras by R. Kadison, 1967”, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 3, 619–624. With a previously unpublished manuscript by Kadison; International Workshop on Operator Algebra and Operator Theory (Linfen, 2001). MR 2014042, DOI 10.1007/s10114-003-0279-x
- Liming Ge, On maximal injective subalgebras of factors, Adv. Math. 118 (1996), no. 1, 34–70. MR 1375951, DOI 10.1006/aima.1996.0017
- L. Ge and R. Kadison, On tensor products for von Neumann algebras, Invent. Math. 123 (1996), no. 3, 453–466. MR 1383957, DOI 10.1007/s002220050036
- ChengJun Hou, On maximal injective subalgebras in a $w\Gamma$ factor, Sci. China Ser. A 51 (2008), no. 11, 2089–2096. MR 2447433, DOI 10.1007/s11425-008-0059-2
- R. Kadison. Problems on von Neumann algebras. Notes of Baton Rouge Conference, unpublished, 1967.
- Richard V. Kadison and John R. Ringrose, Fundamentals of the theory of operator algebras. Vol. II, Graduate Studies in Mathematics, vol. 16, American Mathematical Society, Providence, RI, 1997. Advanced theory; Corrected reprint of the 1986 original. MR 1468230, DOI 10.1090/gsm/016/01
- Sorin Popa, Maximal injective subalgebras in factors associated with free groups, Adv. in Math. 50 (1983), no. 1, 27–48. MR 720738, DOI 10.1016/0001-8708(83)90033-6
- Junhao Shen, Maximal injective subalgebras of tensor products of free group factors, J. Funct. Anal. 240 (2006), no. 2, 334–348. MR 2261686, DOI 10.1016/j.jfa.2006.03.017
- Allan M. Sinclair and Roger R. Smith, Finite von Neumann algebras and masas, London Mathematical Society Lecture Note Series, vol. 351, Cambridge University Press, Cambridge, 2008. MR 2433341, DOI 10.1017/CBO9780511666230
- Şerban Strătilă and László Zsidó, The commutation theorem for tensor products over von Neumann algebras, J. Funct. Anal. 165 (1999), no. 2, 293–346. MR 1698940, DOI 10.1006/jfan.1999.3408
Additional Information
- Mingchu Gao
- Affiliation: Department of Mathematics, Louisiana College, Pineville, Louisiana 71359
- Email: gao@lacollege.edu
- Received by editor(s): March 16, 2009
- Received by editor(s) in revised form: September 20, 2009
- Published electronically: January 7, 2010
- Communicated by: Marius Junge
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 2065-2070
- MSC (2010): Primary 46L10
- DOI: https://doi.org/10.1090/S0002-9939-10-10219-6
- MathSciNet review: 2596043