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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
Published by: Mathematica Scandinavica
Stable URL: http://www.jstor.org/stable/24492947
Page Count: 12
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
ENDOMORPHISMS OF THE ALGEBRA OF ABSOLUTELY CONTINUOUS FUNCTIONS AND OF ALGEBRAS OF ANALYTIC FUNCTIONS
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Abstract

Let π’œπ’ž be the algebra of absolutely continuous functions on the unit circle T. The main result of this paper is that a function Ο„ ∈ π’œπ’ž with Ο„(T) βŠ† T induces an endomorphism of π’œπ’ž by f ↦ f β—‹ Ο„ (f ∈ π’œπ’ž) 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 βˆ‚X denotes the topological boundary of X and #X the number of elements in X). We also discuss endomorphisms of the algebra π’œπ’ž+ = π’œπ’ž ∩ π’œ(Ξ”Μ„)(where π’œ(Ξ”Μ„) is the disc algebra) and of Lipschitz algebras on the closed unit disc Ξ”Μ„.

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