Weak$^{*}$ properties of weighted convolution algebras II

Sandy Grabiner¹

¹ Department of Mathematics Pomona College 610 North College Ave. Claremont, CA 91711, U.S.A.

Studia Mathematica, Tome 198 (2010) no. 1, pp. 53-67

Résumé

We show that if $\phi$ is a continuous homomorphism between weighted convolution algebras on ${\mathbb{R}^{+}},$ then its extension to the corresponding measure algebras is always weak$^{\ast}$ continuous. A key step in the proof is showing that our earlier result that normalized powers of functions in a convolution algebra on ${\mathbb{R}^{+}}$ go to zero weak$^{\ast}$ is also true for most measures in the corresponding measure algebra. For some algebras, we can determine precisely which measures have normalized powers converging to zero weak$^{\ast}$. We also include a variety of applications of weak$^{\ast}$ results, mostly to norm results on ideals and on convergence.

Détail

DOI : 10.4064/sm198-1-3

Keywords: phi continuous homomorphism between weighted convolution algebras mathbb its extension corresponding measure algebras always weak ast continuous key step proof showing earlier result normalized powers functions convolution algebra mathbb zero weak ast measures corresponding measure algebra algebras determine precisely which measures have normalized powers converging zero weak ast include variety applications weak ast results mostly norm results ideals convergence