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On commutative, operator amenable subalgebras of finite von Neumann algebras

Research output: Contribution to journalJournal articlepeer-review

Published
<mark>Journal publication date</mark>05/2013
<mark>Journal</mark>Journal für die reine und angewandte Mathematik (Crelle's Journal)
Volume678
Number of pages22
Pages (from-to)201-222
Publication StatusPublished
Early online date23/02/12
<mark>Original language</mark>English

Abstract

It has been conjectured, motivated in part by old results of Dixmier and Day on bounded Hilbertian representations of amenable groups, that every norm-closed amenable subalgebra of ℬ(ℋ) is automatically similar to an amenable C*-algebra. Results of Curtis and Loy (1995), Gifford (2006), and Marcoux (2008) give some evidence to support this conjecture, but it remains open even for commutative subalgebras.

We present more evidence to support this conjecture, by showing that a closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ℳ must be similar to a selfadjoint-subalgebra. Technical results used include an approximation argument based on Grothendieck's inequality and the Pietsch Domination Theorem, together with an adaptation of a theorem of Gifford to the setting of unbounded operators affiliated to ℳ.