Weak-star properties of homomorphisms from weighted convolution algebras on the half-line

TY - JOUR

T1 - Weak-star properties of homomorphisms from weighted convolution algebras on the half-line

AU - Pedersen, Thomas Vils

PY - 2010/8

Y1 - 2010/8

N2 - Let L1(ω) be the weighted convolution algebra L ω1(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→ L1(ω2), the unique continuous extension Φ:M(ω1)→ M(ω2) 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 A(D) every nonzero homomorphism Φ :L1(ω) → A(D) extends uniquely to a continuous homomorphism Φ̃:M(ω) → H∞(D) which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras Av+ on D̄(including the algebra of absolutely convergent Taylor series on D̄) we prove that every nonzero homomorphism Φ:L1(ω) → Av+ extends uniquely to a continuous homomorphism Φ̄:M(ω) → Av+ which is also continuous with respect to the weak-star topologies.

AB - Let L1(ω) be the weighted convolution algebra L ω1(ℝ+) on ℝ+ with weight ω. Grabiner recently proved that, for a nonzero, continuous homomorphism Φ:L1(ω1)→ L1(ω2), the unique continuous extension Φ:M(ω1)→ M(ω2) 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 A(D) every nonzero homomorphism Φ :L1(ω) → A(D) extends uniquely to a continuous homomorphism Φ̃:M(ω) → H∞(D) which is also continuous with respect to the weak-star topologies. Similarly, for a large class of Beurling algebras Av+ on D̄(including the algebra of absolutely convergent Taylor series on D̄) we prove that every nonzero homomorphism Φ:L1(ω) → Av+ extends uniquely to a continuous homomorphism Φ̄:M(ω) → Av+ which is also continuous with respect to the weak-star topologies.

U2 - 10.1017/S144678871000025X

DO - 10.1017/S144678871000025X

M3 - Journal article

SN - 1446-7887

VL - 89

SP - 75

EP - 90

JO - Journal of the Australian Mathematical Society

JF - Journal of the Australian Mathematical Society

IS - 1

ER -