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Abstract

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.