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Homogeneous algebras on the circle. II. Multipliers, Ditkin conditions

Colin BennettJohn E. Gilbert — 1972

Annales de l'institut Fourier

This paper considers the Lipschitz subalgebras Λ ( α , p , 𝒜 ) of a homogeneous algebra on the circle. Interpolation space theory is used to derive estimates for the multiplier norm on closed primary ideals in Λ ( α , p ; 𝒜 ) , α [ α ] . From these estimates the Ditkin and Analytic Ditkin conditions for Λ ( α , p ; 𝒜 ) follow easily. Thus the well-known theory of (regular) Banach algebras satisfying the Ditkin condition applies to Λ ( α ; , p ; 𝒜 ) as does the theory developed in part I of this series which requires the Analytic Ditkin condition. Examples...

Homogeneous algebras on the circle. I. Ideals of analytic functions

Colin BennettJohn E. Gilbert — 1972

Annales de l'institut Fourier

Let 𝒜 be a homogeneous algebra on the circle and 𝒜 + the closed subalgebra of 𝒜 of functions having analytic extensions into the unit disk D . This paper considers the structure of closed ideals of 𝒜 + under suitable restrictions on the synthesis properties of 𝒜 . In particular, completely characterized are the closed ideals in 𝒜 + whose zero sets meet the circle in a countable set of points. These results contain some previous results of Kahane and Taylor-Williams obtained independently.

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