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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Approximation properties and approximate identities of $ A_{p}(G)$


Author: Tianxuan Miao
Journal: Trans. Amer. Math. Soc. 361 (2009), 1581-1595
MSC (2000): Primary 43A07
Published electronically: October 20, 2008
MathSciNet review: 2457409
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Abstract: For a locally compact group $ G$ and $ 1 < p < \infty $, let $ A_{p}(G)$ be the Figà-Talamanca-Herz algebra. Then the multiplier algebra $ MA_{p}(G)$ of $ A_{p}(G)$ is a dual space. We say that $ A_{p}(G)$ has the approximation property (or simply, AP) in $ MA_{p}(G)$ if there is a net $ \{ u_{\alpha } \}$ in $ A_{p}(G)$ such that $ u_{\alpha }\rightarrow 1$ in the associated $ weak^{*}$ topology. We prove that $ A_{p}(G)$ has the AP in $ MA_{p}(G)$ if and only if there exists a net $ \{ a_{\alpha } \}$ in $ A_{p}(G)$ such that $ \Vert a_{\alpha } a - a\Vert_{A_{p}(G)}\rightarrow 0$ uniformly for $ a$ in any compact subset of $ A_{p}(G)$. Consequently, we have that if $ A_{p}(G)$ has the AP in $ MA_{p}(G)$, then $ A_{p}(G)$ has the approximation property as a Banach space in the sense of Grothendieck for a discrete group $ G$. We also study the relationship between the AP of $ A_{p}(G)$ in $ MA_{p}(G)$ and the weak amenability of $ G$.


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Additional Information

Tianxuan Miao
Affiliation: Department of Mathematics, Lakehead University, Thunder Bay, Ontario, Canada P7B 5E1
Email: tmiao@lakeheadu.ca

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04674-6
PII: S 0002-9947(08)04674-6
Keywords: Amenable groups, multiplier algebra, Herz algebra, approximation property, approximate identity
Received by editor(s): March 2, 2007
Published electronically: October 20, 2008
Additional Notes: This research was supported by an NSERC grant.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.