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Inner functions and the maximal ideal space of L∞(Tn)

Part of: Holomorphic functions of several complex variables Commutative Banach algebras and commutative topological algebras

Published online by Cambridge University Press:  09 April 2009

S. H. Kon
S. H. Kon
Affiliation:
Department of Mathematics University of Malaya Kuala Lumpur 22-11 Malaysia
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Abstract

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Let Un be the unit polydisc in Cn and let Tn be its distinguished boundary. It is shown that a function f ∈ H∞(Un) is inner if and only if ∣f(Φ)∣ = 1 for all Φ in the maximal ideal space of L∞(Tn). This generalizes a result of Csordas.

Keywords

inner functions, H∞(Un), maximal ideal spaceShilov boundarypolydiscL∞(Tn)radial cluster set.

MSC classification

Secondary: 32A30: Other generalizations of function theory of one complex variable (should also be assigned at least one classification number from Section 30) 46J20: Ideals, maximal ideals, boundaries
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 2 , September 1978 , pp. 175 - 178
Copyright
Copyright © Australian Mathematical Society 1978

References

Csordas, G. L. (1973), “A note on the Shilov boundary and the cluster sets of a class of functions in H∞”, Acta Math. Acad. Sci. Hungar. 24, 5–11.CrossRefGoogle Scholar
Hoffman, K. (1962), Banach Spaces of Analytic Functions (Prentice-Hall, Englewood Cliffs, N.J.).Google Scholar
Range, M. (1972), “A small boundary for H∞ on the polydisc”, Proc. Amer. Math. Soc. 32, 253–255.Google Scholar