A generalized Fatou theorem for Banach algebras

Donald Curtis Taylor

Vol. 26 (1968), No. 2, 389–394

DOI: 10.2140/pjm.1968.26.389

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 M_B 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. ∥f∥≦ C
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