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: Suppose that is a weighted convolution algebra on with the weight normalized so that the corresponding space of measures is the dual space of the space of continuous functions. Suppose that is a continuous nonzero homomorphism, where is also a convolution algebra. If is norm dense in , we show that is (relatively) weak dense in , and we identify the norm closure of with the convergence set for a particular semigroup. When is weak continuous it is enough for to be weak dense in . We also give sufficient conditions and characterizations of weak continuity of . In addition, we show that, for all nonzero in , the sequence converges weak to 0. When is regulated, 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