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

Abstract:

Suppose that $L^{1}(\omega )$ is a weighted convolution algebra on $\mathbf {R}^{+}=[0,\infty )$ with the weight $\omega (t)$ normalized so that the corresponding space $M(\omega )$ of measures is the dual space of the space $C_{0}(1/\omega )$ of continuous functions. Suppose that $\phi : L^{1}(\omega )\rightarrow L^1(\omega ’)$ is a continuous nonzero homomorphism, where $L^1(\omega ’)$ is also a convolution algebra. If $L^{1}(\omega )\ast f$ is norm dense in $L^{1}(\omega )$, we show that ${L^{1}(\omega ’)}\ast \phi (f)$ is (relatively) weak$^{\ast }$ dense in $L^1(\omega ’)$, and we identify the norm closure of $L^{1}(\omega ’)\ast \phi (f)$ with the convergence set for a particular semigroup. When $\phi$ is weak$^{\ast }$ continuous it is enough for $L^{1}(\omega )\ast f$ to be weak$^{\ast }$ dense in $L^{1}(\omega )$. We also give sufficient conditions and characterizations of weak$^{\ast }$ continuity of $\phi$. In addition, we show that, for all nonzero $f$ in $L^{1}(\omega )$, the sequence $f^{n}/||f^{n}||$ converges weak$^{\ast }$ to 0. When $\omega$ is regulated, $f^{n+1}/||f^{n}||$ converges to 0 in norm.