Vol. 18, No. 3, 1966

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Embedding theorems for commutative Banach algebras

William George Bade and Philip C. Curtis, Jr.

Vol. 18 (1966), No. 3, 391–409

One knows from the Gelfand theory that every commutative semisimple Banach algebra A containing an identity is a separating subalgebra of the algebra of all complex continuous functions on the space of maximal ideals of A. We shall be concerned in this paper with conditions which when imposed on a separating Banach subalgebra A of C(Ω), Ω a compact Hausdorff space, will guarantee that A = C(Ω). The conditions will take the form of restrictions on either the algebra or the space Ω. For example we prove that if A is an 𝜖-normal Banach subalgebra of C(Ω) then A = C(Ω) if an appropriate boundedness condition holds locally on Ω. If Ω is assumed to be an F space in the sense of Gillman and Henriksen this boundedness assumption is redundant. These results include a recent characterization of Sidon sets in discrete groups due to Rudin and have applications to interpolation problems for bounded analytic functions.

Mathematical Subject Classification
Primary: 46.55
Secondary: 42.58
Received: 20 April 1964
Revised: 6 July 1964
Published: 1 September 1966
William George Bade
Philip C. Curtis, Jr.