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 pr esent the first commutative examples. Centrally placed within the construct ion, the reader may be pleased to notice a reprise of the undergraduate arg ument that shows that a normed space with totally bounded unit ball is fini te-dimensional; we use the same idea (approximate the norm 1 vector x withi n distance eta by a "good" vector y(1); then approximate (x - y(1))/eta wit hin distance eta by a "good" vector y(2), thus approximating x within dista nce eta (2) by y(1) + etay(2), and so on) to go from eta = 9/10 in Lemma 1. 5 to arbitrarily small eta 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 eta small at the start will not do.