The Bidual of a Radical Operator algebra can be Semisimple

Abstract

The paper of S. Gulick [Sidney (Denny) L. Gulick, Commutativity and ideals in the biduals of topological algebras, Pacific J. Math 18 No. 1, 1966] contains some good mathematics, but it also contains an error. It claims that for a Banach algebra A, the intersection of the Jacobson radical of A** with A is precisely the radical of A (this is claimed for either of the Arens products on A** - in itself a reasonable claim, because A is always contained in the topological centre of A**, so a fixed a in A lies in the radical of A** with the first Arens product, if and only if it lies in the radical of A** when that Banach space is given the second Arens product, if and only if ab is quasinilpotent for every b in A**). In this paper we begin with a simple counterexample to that claim, in which A is a radical operator algebra, but not every element of A lies in the radical of A**. We then develope a more complicated example A which, once again, is a radical operator algebra, but A** is semisimple. So rad A** $\cap$ A is zero, but rad A=A. We conclude by examining the uses Gulick's paper has been put to since 1966 (at least 8 subsequent papers refer to it), and we find that most authors have used the correct material from that paper, and avoided using the wrong result. We reckon, then, that we are not the first to suspect that the result rad A** $\cap$ A=rad A was wrong; but we believe we are the first to provide "neat" counterexamples as described.