The L^p theory of standard homomorphisms

Fereidoun Ghahramani and Sandy Grabiner

Vol. 168 (1995), No. 1, 49–60

DOI: 10.2140/pjm.1995.168.49

Abstract

Suppose that ϕ : L¹(ω₁) → L¹(ω₂) is a continuous nonzero homomorphism between weighted convolution algebras on R⁺, and let ϕ also designate the extension of this map to the corresponding measure algebras M(ω₁) and M(ω₂). For 1 < p < ∞, we prove: (a) the semigroup μ_t = ϕ(δ_t) acts as a strongly continuous semigroup on L^p(ω₂); (b) Whenever L¹(ω₁) ∗ f is dense in L¹(ω₁), then L^p(ω₂) ∗ ϕ(f) is dense in L^p(ω₂); (c) Each h in L^p(ω₂) can be factored as h = ϕ(f) ∗g; (d) ϕ is continuous from the strong operator topology of M(ω₁) acting on L¹(ω₁) to the strong operator topology of M(ω₂) acting on L^p(ω₂).

Mathematical Subject Classification 2000

Primary: 43A22

Secondary: 43A15

Milestones

Received: 21 September 1992

Revised: 21 March 1993

Published: 1 March 1995