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CO-REPRESENTATIONS OF HOPF–VON NEUMANN ALGEBRAS ON OPERATOR SPACES OTHER THAN COLUMN HILBERT SPACE

Part of: Selfadjoint operator algebras Linear spaces and algebras of operators

Published online by Cambridge University Press:  22 June 2010

VOLKER RUNDE
VOLKER RUNDE*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 (email: vrunde@ualberta.ca)
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Abstract

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Recently, Daws introduced a notion of co-representation of abelian Hopf–von Neumann algebras on general reflexive Banach spaces. In this note, we show that this notion cannot be extended beyond subhomogeneous Hopf–von Neumann algebras. The key is our observation that, for a von Neumann algebra 𝔐 and a reflexive operator space E, the normal spatial tensor product is a Banach algebra if and only if 𝔐 is subhomogeneous or E is completely isomorphic to column Hilbert space.

Keywords

Hopf–von Neumann algebraco-representationreflexive operator spaceoperator algebracolumn Hilbert space

MSC classification

Secondary: 47L30: Abstract operator algebras on Hilbert spaces 46L99: None of the above, but in this section 47L25: Operator spaces (= matricially normed spaces)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 2 , October 2010 , pp. 205 - 210
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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