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WEAK-STAR PROPERTIES OF HOMOMORPHISMS FROM WEIGHTED CONVOLUTION ALGEBRAS ON THE HALF-LINE
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 89 / Issue 1 / August 2010
- Published online by Cambridge University Press:
- 20 July 2010, pp. 75-90
- Print publication:
- August 2010
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- Article
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Let L1(ω) be the weighted convolution algebra L1ω(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→L1(ω2), the unique continuous extension
to a homomorphism between the corresponding weighted measure algebras on ℝ+ is also continuous with respect to the weak-star topologies on these algebras. In this paper we investigate whether similar results hold for homomorphisms from L1(ω) into other commutative Banach algebras. In particular, we prove that for the disc algebra 
every nonzero homomorphism 
extends uniquely to a continuous homomorphism 
which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras A+v on 
(including the algebra of absolutely convergent Taylor series on ) we prove that every nonzero homomorphism Φ:L1(ω)→A+v extends uniquely to a continuous homomorphism 
which is also continuous with respect to the weak-star topologies.
to a homomorphism between the corresponding weighted measure algebras on ℝ
every nonzero homomorphism 
extends uniquely to a continuous homomorphism 
which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras 
(including the algebra of absolutely convergent Taylor series on 
which is also continuous with respect to the weak-star topologies.