Reflexivity and hyperreflexivity of bounded $N$-cocycles from group algebras
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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.References
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Additional Information
- Ebrahim Samei
- Affiliation: Department of Mathematics and Statistics, University of Saskatchewan, 106 Wiggins Road, Saskatoon, SK, Canada
- Email: samei@math.usask.ca
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 163-176
- MSC (2010): Primary 47B47, 43A20
- DOI: https://doi.org/10.1090/S0002-9939-2010-10454-9
- MathSciNet review: 2729080