Transactions of the American Mathematical Society

Author(s): Allan M. Sinclair; Roger R. Smith
Journal: Trans. Amer. Math. Soc. 355 (2003), 465-475.
MSC (2000): Primary 46L10, 46L09
Posted: October 9, 2002
Retrieve article in: PDF DVI PostScript

Abstract: The Laplacian (or radial) masa in a free group factor is generated by the sum of the generators and their inverses. We show that such a masa $\mathcal{B}$ is strongly singular and has Popa invariant $\delta(\mathcal{B}) = 1$ . This is achieved by proving that the conditional expectation $\mathbb{E} _{\mathcal{B}}$ onto $\mathcal{B}$ is an asymptotic homomorphism. We also obtain similar results for the free product of discrete groups, each of which contains an element of infinite order.

1.: F. Boca and F. Radulescu, Singularity of radial subalgebras in ${\rm II}_1$ factors associated with free products of groups, J. Funct. Anal., 103 (1992), 138-159. MR 93d:46103
2.: J. Dixmier, Sous-anneaux abéliens maximaux dans les facteurs de type fini, Ann. Math., 59 (1954), 279-286. MR 15:539b
3.: A. Figà-Talamanca and M. Picardello, Harmonic analysis on free groups, Lecture Notes in Pure and Applied Mathematics, 87, Marcel Dekker, New York, 1983. MR 85j:43001
4.: S. Popa, Singular maximal abelian -subalgebras in continuous von Neumann algebras, J. Funct. Anal., 50 (1983), 151-166. MR 84e:46065
5.: S. Popa, Notes on Cartan subalgebras in type ${II}_1$ factors, Math. Scand., 57 (1985), 171-188. MR 87f:46114
6.: S. Popa, On the distance between masas in type ${II}_1$ factors, Math. Phys. in Math. and Phys. (Siena, 2000), Fields Inst. Commun., vol. 30, Amer. Math. Soc., Providence, RI, 2001, pp. 321-324.
7.: L. Pukánszky, On maximal abelian subrings of factors of type ${\rm II}_{1}$ , Canad. J. Math., 12 (1960), 289-296. MR 22:3992
8.: 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 84a:22017
9.: F. Radulescu, Singularity of the radial subalgebra of ${\mathcal{L}}(F_N)$ and the Pukánszky invariant, Pacific J. Math., 151 (1991), 297-306. MR 93b:46120
10.: A. M. Sinclair and R. R. Smith, Strongly singular masas in type ${II}_1$ factors, Geom. Funct. Anal., 12 (2002), 199-216.
11.: 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 96m:46119

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 46L10, 46L09

Allan M. Sinclair
Affiliation: School of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland
Email: allan@maths.ed.ac.uk

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS