Smooth Banach algebras

Abstract

Introduction. The purpose of this note is to prove the theorem that follows. This theorem generalizes results of L. Ingelstam [2] and M. F. Smiley [4] for Hilbert algebras. It also provides a simpler proof of Corollary which was obtained by F. F. Bonsall and J. Duncan [1]. (This result was obtained before I was aware of the work of Bonsall and Duncan.) Let G(x, y) (G+(x, y), G_(x, y)) denote the limit as t approaches zero (from the right, from the left) of (||x+tyl -||x|)/t. For any of the properties of G used in the proof see Koethe [3].