Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-25T04:18:20.787Z Has data issue: false hasContentIssue false
  • English
  • Français

Boundaries for polydisc algebras in infinite dimensions

Published online by Cambridge University Press:  24 October 2008

J. Globevnik
J. Globevnik
Affiliation:
University of Ljubljana
Get access Rights & Permissions [Opens in a new window]

Abstract

Let AB be the algebra of all bounded continuous functions on the closed unit ball B of c0, analytic on the open unit ball, with sup norm, and let AU be the sub-algebra of AB of those functions which are uniformly continuous on B. Call a set S ⊂ B a boundary of AB (AU) if

for every f ∈ AB (f ∈AU, respectively). In the paper we study the boundaries of AB and AU. We give a complete description of the boundaries of AU and present some necessary and some sufficient conditions for a set to be a boundary of AB. We also give some examples of boundaries.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 85 , Issue 2 , March 1979 , pp. 291 - 303
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Bishop, E.A minimal boundary for function algebras. Pacific J. Math. 9 (1959), 629–642.CrossRefGoogle Scholar
(2)Globevnik, J.On interpolation by analytic maps in infinite dimensions. Math. Proc. Cambridge Philos. Soc. 83 (1978), 243–252.CrossRefGoogle Scholar
(3)Harris, L. A.Schwarz's lemma in normed linear spaces. Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1014–1017.CrossRefGoogle ScholarPubMed
(4)Hille, E. and Phillips, R. S.Functional analysis and semi-groups. Amer. Math. Soc. Colloq. Publ. 31 (1957).Google Scholar
(5)Hoffman, K.Banach spaces of analytic functions (Prentice–Hall, 1962).Google Scholar
(6)Leibowitz, G. M.Lectures on complex function algebras (Scott, Foresman, 1970).Google Scholar
(7)Nachbin, L.Topology on spaces of holomorphic mappings. Ergebn. Math. (Springer, 1969).CrossRefGoogle Scholar
(8)Polya, G. and Szegö, G.Aufgaben und Lehrsätze aus der Analysis Bd. 1 (Springer, 1954).CrossRefGoogle Scholar
(9)Rudin, W.Function theory in polydiscs (Benjamin, 1969).Google Scholar
(10)Rudin, W.Real and complex analysis (McGraw-Hill, 1970).Google Scholar
(11)Stout, E. L.The theory of uniform algebras (Bogden and Quigley, 1971).Google Scholar