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Reflexivity and hyperreflexivity of bounded $ N$-cocycles from group algebras

Author: Ebrahim Samei
Journal: Proc. Amer. Math. Soc. 139 (2011), 163-176
MSC (2010): Primary 47B47, 43A20
Published electronically: June 30, 2010
MathSciNet review: 2729080
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Abstract: We introduce the concept of reflexivity for bounded $ n$-linear maps and investigate the reflexivity of $ \mathcal{Z}^n(L^1(G),X)$, the space of bounded $ n$-cocycles from $ L^1(G)^{(n)}$ into $ X$, where $ L^1(G)$ is the group algebra of a locally compact group $ G$ and $ X$ is a Banach $ L^1(G)$-bimodule. We show that $ \mathcal{Z}^n(L^1(G),X)$ is reflexive for a large class of groups including groups with polynomial growth, IN-groups, maximally almost periodic groups, and totally disconnected groups. If, in addition, $ G$ is amenable and $ X$ is the dual of an essential Banach $ L^1(G)$-bimodule, then we show that $ \mathcal{Z}^1(L^1(G),X)$ satisfies a stronger property, namely hyperreflexivity. This, in particular, implies that $ \mathcal{Z}^1(L^1(G),L^1(G))$ is hyperreflexive.

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Additional Information

Ebrahim Samei
Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, Canada

Keywords: Reflexivity, hyperreflexivity, $n$-cocycles, $n$-hyperlocal maps, the derivation space, group algebras, groups with polynomial growth
Received by editor(s): January 3, 2010
Received by editor(s) in revised form: February 20, 2010
Published electronically: June 30, 2010
Additional Notes: This work was partially supported by an NSERC Discovery Grant
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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