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Homomorphisms of the algebra of locally integrable functions on the half line

Part of: Commutative Banach algebras and commutative topological algebras Abstract harmonic analysis

Published online by Cambridge University Press:  09 April 2009

Sandy Grabiner
Sandy Grabiner
Affiliation:
Department of Mathematics, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA e-mail: sgrabiner@pomona.edu
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Abstract

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Let φ be a continuous nonzero homomorphism of the convolution algebra L1loc(R+) and also the unique extension of this homomorphism to Mloc(R+). We show that the map φis continuous in the weak* and strong opertor topologies on Mloc, considered as the dual space of Cc(R+) and as the multiplier algebra of L1loc. Analogous results are proved for homomorphism from L1 [0, a) to L1 [0, b). For each convolution algebra L1 (ω1), φ restricts to a continuous homomorphism from some L1 (ω1) to some L1 (ω2), and, for each sufficiently large L1 (ω2), φ restricts to a continuous homomorphism from some L1 (ω1) to L1 (ω2). We also determine which continuous homomorphisms between weighted convolution algebras extend to homomorphisms of L1loc. We also prove results on convergent nets, continuous semigroups, and bounded sets in Mloc that we need in our study of homomorphisms.

MSC classification

Secondary: 43A22: Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 46J40: Structure, classification of commutative topological algebras 46J45: Radical Banach algebras 43A20: $L^1$-algebras on groups, semigroups, etc. 43A10: Measure algebras on groups, semigroups, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 253 - 278
Copyright
Copyright © Australian Mathematical Society 2006

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