Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Decomposition of $B(G)$
HTML articles powered by AMS MathViewer

by Tianxuan Miao PDF
Trans. Amer. Math. Soc. 351 (1999), 4675-4692 Request permission

Abstract:

For any locally compact group $G$, let $A(G)$ and $B(G)$ be the Fourier and the Fourier-Stieltjes algebras of $G$, respectively. $B(G)$ is decomposed as a direct sum of $A(G)$ and $B^{s}(G)$, where $B^{s}(G)$ is a subspace of $B(G)$ consisting of all elements $b\in B(G)$ that satisfy the property: for any $\epsilon > 0$ and any compact subset $K\subset G$, there is an $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$ such that $\vert \langle f, b \rangle \vert > \Vert b\Vert - \epsilon .$ $A(G)$ is characterized by the following: an element $b\in B(G)$ is in $A(G)$ if and only if, for any $\epsilon > 0,$ there is a compact subset $K\subset G$ such that $\vert \langle f, b \rangle \vert < \epsilon$ for all $f\in L^{1}(G)$ with $\Vert f\Vert _{C^{*}(G)} \le 1$ and $supp(f) \subset K^{c}$. Note that we do not assume the amenability of $G$. Consequently, we have $\Vert 1 + a\Vert = 1 + \Vert a\Vert$ for all $a\in A(G)$ if $G$ is noncompact. We will apply this characterization of $B^{s}(G)$ to investigate the general properties of $B^{s}(G)$ and we will see that $B^{s}(G)$ is not a subalgebra of $B(G)$ even for abelian locally compact groups. If $G$ is an amenable locally compact group, then $B^{s}(G)$ is the subspace of $B(G)$ consisting of all elements $b\in B(G)$ with the property that for any compact subset $K\subseteq G$, $\Vert b\Vert = \sup \{ \Vert a b\Vert : a\in A(G), \; supp(a) \subseteq K^{c} \; \text { and } \; \Vert a\Vert \le 1 \}$.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 43A07
  • Retrieve articles in all journals with MSC (1991): 43A07
Additional Information
  • Tianxuan Miao
  • Affiliation: Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ontario P7E 5E1 Canada
  • Email: tmiao@thunder.lakeheadu.ca
  • Received by editor(s): April 29, 1997
  • Published electronically: July 20, 1999
  • Additional Notes: This research is supported by an NSERC grant
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 351 (1999), 4675-4692
  • MSC (1991): Primary 43A07
  • DOI: https://doi.org/10.1090/S0002-9947-99-02328-4
  • MathSciNet review: 1608490