Commutative, radical amenable Banach algebras

There has been a considerable search for radical, amenable Banach algebras. Noncommutative examples were finally found by Volker Runde [R]; here we present the first commutative examples. Centrally placed within the construction, the reader may be pleased to notice a reprise of the undergraduate argument that shows that a normed space with totally bounded unit ball is finite-dimensional; we use the same idea (approximate the norm 1 vector x within distance η by a "good" vector ${\displaystyle y_1}$; then approximate ${\displaystyle \frac{x-y_1}{\eta }}$ within distance η by a "good" vector ${\displaystyle y_2}$, thus approximating x within distance ${\displaystyle {\eta }^2}$ by ${\displaystyle y_1+\eta y_2}$, and so on) to go from η=9/10 in Lemma 1.5 to arbitrarily small η in Lemma 2.1. This is not an arbitrary decision on the part of the author; it really is forced on him by the nature of the construction, see e.g. (6.1) for a place where η small at the start will not do.