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# ENDOMORPHISMS OF THE ALGEBRA OF ABSOLUTELY CONTINUOUS FUNCTIONS AND OF ALGEBRAS OF ANALYTIC FUNCTIONS

THOMAS VILS PEDERSEN
Mathematica Scandinavica
Vol. 82, No. 1 (1998), pp. 89-100
Stable URL: http://www.jstor.org/stable/24492947
Page Count: 12
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## Abstract

Let 𝒜𝒞 be the algebra of absolutely continuous functions on the unit circle T. The main result of this paper is that a function τ ∈ 𝒜𝒞 with τ(T) ⊆ T induces an endomorphism of 𝒜𝒞 by f ↦ f ○ τ (f ∈ 𝒜𝒞) if and only if $\underset{\mathrm{t}\in \mathrm{T}}{\mathrm{sup}}\mathrm{\#}(\partial ({\mathrm{\tau }}^{-1}\left(\mathrm{t}\right)\left)\right)<\mathrm{\infty }$ (where ∂X denotes the topological boundary of X and #X the number of elements in X). We also discuss endomorphisms of the algebra 𝒜𝒞+ = 𝒜𝒞 ∩ 𝒜(Δ̄)(where 𝒜(Δ̄) is the disc algebra) and of Lipschitz algebras on the closed unit disc Δ̄.

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