Journal Article

AMENABILITY AND MODULES FOR ARENS PRODUCT ALGEBRAS

Ross Stokke

in The Quarterly Journal of Mathematics

Volume 66, issue 1, pages 295-321
Published in print March 2015 | ISSN: 0033-5606
Published online June 2014 | e-ISSN: 1464-3847 | DOI: https://dx.doi.org/10.1093/qmath/hau018

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Given a Banach algebra A, we introduce the notion of a left dual Banach algebra (LDBA) over A, and we establish that every LDBA over A is a left Arens product algebra over A. This can be viewed as a Banach algebraic version of the fact that every semigroup compactification is a Gelfand compactification. We show how A-module operations can be extended to obtain module operations for left Arens product algebras over A that satisfy attractive w*-continuity properties. We introduce a notion of left Connes amenability for LDBAs, and show that the amenability of a locally compact group G is equivalent to left Connes amenability of either the bidual L1(G)** of its group algebra L1(G), or the dual LUC(G)* where LUC(G) is the space of left uniformly continuous functions on G.

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Subjects: Pure Mathematics