Cohen–Host type idempotent theorems for representations on Banach spaces and applications to Figà-Talamanca–Herz algebras
- 作者：Volker Runde
- 关键词：Amenability ; Bounded approximate identity ; Coset ring ; Figà ; -Talamanca– ; Herz algebra ; Locally compact group ; Smooth Banach space ; Ultrapower ; Uniform convexity ; Uniformly bounded representation
- 刊名：Journal of Mathematical Analysis and Applications
- 出版年：2007
- 期刊代码：76_0022247x
- 类别：mt
- 出版时间：1 May 2007
- 卷：329
- 期：1
- 页码：736-751
- 文件大小：192 K
摘要
Let G be a locally compact group, and let e="mml1"> denote the ring of subsets of G generated by the left cosets of open subsets of G. The Cohen–Host idempotent theorem asserts that a set lies in e="mml2"> if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We apply our Cohen–Host type theorems to the study of the Figà-Talamanca–Herz algebras e="mml3">e="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4KJTNTW-C&_mathId=mml3&_user=10&_cdi=6894&_rdoc=54&_acct=C000050221&_version=1&_userid=10&md5=5720030e5b432857767fd8c62943620a" title="Click to view the MathML source">Ap(G) with e="mml4">e="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4KJTNTW-C&_mathId=mml4&_user=10&_cdi=6894&_rdoc=54&_acct=C000050221&_version=1&_userid=10&md5=eaec80c12b9a60141038782f58a419f6" title="Click to view the MathML source">p(1,ȡe;). For arbitrary G, we characterize those closed ideals of e="mml5">e="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4KJTNTW-C&_mathId=mml5&_user=10&_cdi=6894&_rdoc=54&_acct=C000050221&_version=1&_userid=10&md5=9404b9bdb26decb16c622231f8275367" title="Click to view the MathML source">Ap(G) that have an approximate identity bounded by 1 in terms of their hulls. Furthermore, we characterize those G such that e="mml6">e="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4KJTNTW-C&_mathId=mml6&_user=10&_cdi=6894&_rdoc=54&_acct=C000050221&_version=1&_userid=10&md5=0f39679d443800702ac925f20f825e2a" title="Click to view the MathML source">Ap(G) is 1-amenable for some—and, equivalently, for all—e="mml7">e="text-decoration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6WK2-4KJTNTW-C&_mathId=mml7&_user=10&_cdi=6894&_rdoc=54&_acct=C000050221&_version=1&_userid=10&md5=f399db64bc692d637f3bfab3f941faab" title="Click to view the MathML source">p(1,ȡe;): these are precisely the abelian groups.