Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Laplacian subalgebra of $ \mathcal{L}(\mathbb{F}_N)^{\overline{\otimes}_k}$ is a strongly singular masa

Author: Teodor Stefan Bîldea
Journal: Proc. Amer. Math. Soc. 135 (2007), 823-831
MSC (2000): Primary 46L10; Secondary 20E05
Published electronically: August 31, 2006
MathSciNet review: 2262878
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the $ k$-folded tensor product of the free group factor $ \mathcal{L}(\mathbb{F}_N)$ with itself ($ N\ge 2$). The notions of strongly singular masas in type $ {\rm II_1}$ factors and that of asymptotic homomorphism were introduced by A. Sinclair and R. Smith. One of their first examples was the Laplacian subalgebra of the free group factor, generated by the sum of words of length 1 in $ \mathbb{F}_N$. This subalgebra was known to be a singular masa. Using the results of A. Sinclair and R. Smith, we show that the unique trace-preserving conditional expectation onto the Laplacian subalgebra of $ \mathcal{L}(\mathbb{F}_N)^{\overline{\otimes}_k}$ is an asymptotic homomorphism, and hence the Laplacian subalgebra is a strongly singular masa for every $ k\ge 1$.

References [Enhancements On Off] (What's this?)

  • 1. F. Radulescu, Singularity of the radial subalgebra of $ \mathcal{{L}}(\mathbb{{F}}_{{N}})$ and the Pukánsky invariant, Pacific J. Math. 151, no. 2. MR 1132391 (93b:46120)
  • 2. F. Boca and F. Radulescu, Singularity of radial subalgebras in type $ {{\rm {II}}}_1$ factors asociated with free products of groups, J. Funct. Anal. 103 (1992), no. 1, 138-159. MR 1144687 (93d:46103)
  • 3. J. Dixmier, Sous-anneaux abeliens maximaux dans les facteurs de type fini, Ann. Math. 59 (1954), 279-286. MR 0059486 (15:539b)
  • 4. S. Popa, Singular maximal abelian $ \ast$-subalgebras in continuous von Neumann algebras, J. Funct. Anal. 103 (1983), 151-166. MR 0693226 (84e:46065)
  • 5. -, Notes on Cartan subalgebras in type $ {{\rm {II}}}_1$ factors, Math. Scand. 57 (1985), 171-188. MR 0815434 (87f:46114)
  • 6. T. Pytlik, Radial functions on free groups and a decomposition of the regular representation into irreducible components, J. Reine Angew. Math. 326 (1981), 124-135. MR 0622348 (84a:22017)
  • 7. G. Robertson, A.M. Sinclair, and R.R. Smith, Strong singularity for subalgebras of finite factors, Intern. J. Math. 14, no. 3. MR 1978447 (2004c:22007)
  • 8. A.M. Sinclair and R.R. Smith, The Laplacian masa in a free group factor, Trans. Amer. Math. Soc. 355, no. 2. MR 1932708 (2003k:46088)
  • 9. -, Strongly singular masas in type $ {{\rm {II}}}_1$ factors, Geom. Funct. Anal. 12, no. 1. MR 1904563 (2003i:46066)
  • 10. D. Voiculescu, The analogues of entropy and of Fisher's information measure in free probability theory, III: The absence of Cartan subalgebras, Geom. Funct. Anal. 6 (1996), 172-199. MR 1371236 (96m:46119)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46L10, 20E05

Retrieve articles in all journals with MSC (2000): 46L10, 20E05

Additional Information

Teodor Stefan Bîldea
Affiliation: Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, Iowa 52242
Address at time of publication: Computational Biomedicine Lab, Texas Learning and Computation Center, University of Houston, 4800 Calhoun Road, Room 218 Philip Guthrie Hoffman Hall (PGH), Houston, Texas 77204

Keywords: Von Neumann algebra, free group, strongly singular masa, asymptotic homomorphism
Received by editor(s): March 23, 2005
Received by editor(s) in revised form: October 17, 2005
Published electronically: August 31, 2006
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society