Normalisers of irreducible subfactors

Smith, R., White, S. and Wiggins, A. (2009) Normalisers of irreducible subfactors. Journal of Mathematical Analysis and Applications, 352(2), pp. 684-695. (doi: 10.1016/j.jmaa.2008.11.019)

Abstract

We consider normalizers of an infinite index irreducible inclusion Nsubset of or equal toM of II1 factors. Unlike the finite index setting, an inclusion uNu*subset of or equal toN can be strict, forcing us to also investigate the semigroup of one-sided normalizers. We relate these one-sided normalizers of N in M to projections in the basic construction and show that every trace one projection in the relative commutant N′∩left angle bracketM,eNright-pointing angle bracket is of the form u*eNu for some unitary uset membership, variantM with uNu*subset of or equal toN generalizing the finite index situation considered by Pimsner and Popa. We use this to show that each normalizer of a tensor product of irreducible subfactors is a tensor product of normalizers modulo a unitary. We also examine normalizers of infinite index irreducible subfactors arising from subgroup–group inclusions Hsubset of or equal toG. Here the one-sided normalizers arise from appropriate group elements modulo a unitary from L(H). We are also able to identify the finite trace L(H)-bimodules in ℓ2(G) as double cosets which are also finite unions of left cosets.

• Item Type: Articles
• Keywords: Normalizers, irreducible subfactors, infinite index subfactors, group–subgroup inclusions, tensor products, finite factors
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: White, Professor Stuart
• Authors: Smith, R., White, S., and Wiggins, A.
• Subjects: Q Science > QA Mathematics
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal of Mathematical Analysis and Applications
• Journal Abbr.: J. Math. Anal. Appl.
• Publisher: Elsevier
• ISSN: 0022-247X
• ISSN (Online): 1096-0813
• Published Online: 13 November 2008
• Copyright Holders: Copyright © 2009 Elsevier
• First Published: First published in Journal of Mathematical Analysis and Applications 352(2):684-695
• Publisher Policy: Reproduced in accordance with the copyright policy of the publisher.