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


Author: Sandy Grabiner
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1675-1684
MSC (2000): Primary 43A10, 43A20, 43A22, 46J45
DOI: https://doi.org/10.1090/S0002-9939-04-07385-X
Published electronically: January 12, 2004
MathSciNet review: 2051128
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Abstract | References | Similar Articles | Additional Information

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 \ensuremath{L^{1}(\omega')} $ is a continuous nonzero homomorphism, where \ensuremath{L^{1}(\omega')} is also a convolution algebra. If $L^{1}(\omega)\ast f$ is norm dense in $L^{1}(\omega)$, we show that $\ensuremath{L^{1}(\omega')}\ast\phi (f)$ is (relatively) weak$^{\ast}$ dense in \ensuremath{L^{1}(\omega')}, and we identify the norm closure of $\ensuremath{L^{1}(\omega')}\ast\phi (f)$ 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 \ensuremath{L^{1}(\omega )}, the sequence $f^{n}/\vert\vert f^{n}\vert\vert$ converges weak$^{\ast}$ to 0. When $\omega$ is regulated, $f^{n+1}/\vert\vert f^{n}\vert\vert$ converges to 0 in norm.


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Additional Information

Sandy Grabiner
Affiliation: Department of Mathematics, Pomona College, Claremont, California 91711

DOI: https://doi.org/10.1090/S0002-9939-04-07385-X
Received by editor(s): October 9, 2002
Published electronically: January 12, 2004
Additional Notes: The research for this paper was done while the author enjoyed the gracious hospitality of the Australian National University in Canberra
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2004 American Mathematical Society

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