Vol. 26, No. 2, 1968

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A generalized Fatou theorem for Banach algebras

Donald Curtis Taylor

Vol. 26 (1968), No. 2, 389–394
Abstract

Let B denote a commutative, semisimple Banach algebra with unit and let I be a fixed closed ideal in B. In the maximal ideal space MB of B, fix a compact set X and put E = X h(I), where h(I) is the hull of I. The main result of this note is the following

Theorem 1.1. Let I have an approximate unit that is uniformly bounded by the constant C and let g be a nonnegative continuous function on X of sup-norm < 1 that vanishes on E. If h is an element of I and δ > 0, then there exists an element f in I such that

  1. fC
  2. Ref(x) g(x) + |Imf(x)| (x X)
  3. fh h< δ.

Mathematical Subject Classification
Primary: 46.55
Milestones
Received: 27 November 1967
Published: 1 August 1968
Authors
Donald Curtis Taylor