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Closed Ideals in the Ball Algebra

Closed Ideals in the Ball Algebra HAKA N HEDENMALM Section 0 n QO Let B be the open unit ball of C , and let H (B^) be the algebra of bounded analytic functions on B , endowed with the uniform norm on B . The ball algebra is n n the space A(B ) = C(B ) n //°°(Z? ), also given the uniform norm on B . We will write n n ra n S = dB , D = B and T = S For a collection J " of functions in A(B ), associate the n n lt v n zero set f(z) = 0 for all fe&}, and if £ c B , introduce the closed ideal Jf(E) = {feA(B ):f=O on E). In Section 2, we will show every closed ideal / in A(B ) has the form where [I] , denotes the weak-star closure of / in //°°(/? ). However, we have not yet wi n said what the weak-star topology on //°°(/? ) is. The space L™(S ) is the dual space ra n of ^(S ,,), so it has a weak-star topology. One can think of //°°(# ) as a subspace of L (S ) via http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Closed Ideals in the Ball Algebra

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References (6)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/21.5.469
Publisher site
See Article on Publisher Site

Abstract

HAKA N HEDENMALM Section 0 n QO Let B be the open unit ball of C , and let H (B^) be the algebra of bounded analytic functions on B , endowed with the uniform norm on B . The ball algebra is n n the space A(B ) = C(B ) n //°°(Z? ), also given the uniform norm on B . We will write n n ra n S = dB , D = B and T = S For a collection J " of functions in A(B ), associate the n n lt v n zero set f(z) = 0 for all fe&}, and if £ c B , introduce the closed ideal Jf(E) = {feA(B ):f=O on E). In Section 2, we will show every closed ideal / in A(B ) has the form where [I] , denotes the weak-star closure of / in //°°(/? ). However, we have not yet wi n said what the weak-star topology on //°°(/? ) is. The space L™(S ) is the dual space ra n of ^(S ,,), so it has a weak-star topology. One can think of //°°(# ) as a subspace of L (S ) via

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Sep 1, 1989

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