Good weights for weighted convolution algebras

Volume 91 / 2010

Sandy Grabiner Banach Center Publications 91 (2010), 179-189 MSC: Primary 43A20; Secondary 43A10. DOI: 10.4064/bc91-0-10

Abstract

Weighted convolution algebras $L^{1}(\omega )$ on ${\bf R}^{+} = [0,\infty )$ have been studied for many years. At first results were proved for continuous weights; and then it was shown that all such results would also hold for properly normalized right continuous weights. For measurable weights, it was shown that one could construct a properly normalized right continuous weight $\omega '$ with $L^{1}(\omega ') = L^{1}(\omega )$ with an equivalent norm. Thus all algebraic and norm-topology results remained true for measurable weights. We now show that, with careful definitions, the same is true for the weak$^{\ast}$ topology on the space of measures that is the dual of the space of continuous functions $C_{0}(1/\omega )$. We give the new result and a survey of the older results, with several improved statements and/or proofs of theorems.

Authors

  • Sandy GrabinerDepartment of Mathematics
    Pomona College
    610 N. College Avenue
    Claremont, CA 91711, U.S.A.
    e-mail

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