PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 92, Number 3, November 1984 REGULARITY OF BANACH ALGEBRAS GENERATED BY ANALYTIC SEMIGROUPS SATISFYING SOME GROWTH CONDITIONS J. ESTERLE AND J. E. GALE1 ABSTRACT. We show that if a commutative complex Banach algebra A is generated by a nonzero analytic semigroup (a'jReOO satisfying f+°° log+||a1+lí|| J J-oo 1 "I" t then A is regular in Shilov's sense. 1. Introduction. Recall that a commutative (nonradical) complex Banach algebra A is said to be regular in Shilov's sense, or regular for short, if, given ao G A and a neighbourhood U of an in A there exists b G A with ao(b) ^ 0 and a(b) = 0 for every a ^ U. (Here we denote by A the carrier space of A equipped with the Gelfand topology.) The classical proofs of the Wiener tauberian theorem for the convolution algebra LX(R) use the fact that L1(R) is regular. The Wiener tauberian theorem extends to the weighted algebras LX(R, e^')) where <j> is a nonnegative, measurable function on R satisfying <j>(s + t) < (j>(s)+ tj>(t) (s, t G R) and (the first condition just ensures that L1(R, e^O) is stable under convolutions), and Beurling's original proof of this result [1] is also based upon the regularity of L1(R, e^')). There are some noncommutative extensions of Wiener's tauberian theorem to the group algebras L1(G) where G is, for example, a nilpotent Lie group. In that case it is true that if 7 is a two-sided ideal of L1(G) such that I t¿ Ll(G) then the quotient algebra L1(G)/I is not radical (in other terms Ll{G) possesses the "weak Wiener property" in the sense of Leptin [7]). This fact is proved by Hulanicki in [5] using the regularity of the commutative subalgebra of LX(G) generated by the "heat semigroup" (see [7] for a discussion of noncommutative one-sided or two-sided extensions of the tauberian theorem). In [4] the first author gives a proof of the Wiener tauberian theorem for LX(R) based upon an application of the Ahlfors-Heins theorem [2, Theorem 7-2-6] to the heat semigroup. Dales and Hayman give in [3] a proof of the tauberian theorem for some Beurling algebras which also involves the growth on vertical lines of the heat Received by the editors December 12, 1982. 1980 Mathematics Subject Classification. Primary 46J30. Key words and phrases. Regular Banach algebra, analytic semigroup. xThe research of the second author was supported by a Beca del Plan de Formación del Personal Investigador, Spain, in the University of Bordeaux, France. ©1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 377 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use