Vol. 273, No. 2, 2015

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The bidual of a radical operator algebra can be semisimple

Charles John Read

Vol. 273 (2015), No. 2, 443–460
Abstract

The paper of Sidney (Denny) L. Gulick (“Commutativity and ideals in the biduals of topological algebras”, Pacific J. Math. 18, 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 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 develop a more complicated example A, which, once again, is a radical operator algebra, but A is semisimple. So radAA is zero, but radA = 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 radA A = radA was wrong; but we believe we are the first to provide “neat” counterexamples as described.

Keywords
radical Banach algebra, bidual operator algebra
Mathematical Subject Classification 2010
Primary: 46H05, 47L50
Milestones
Received: 5 March 2014
Accepted: 2 June 2014
Published: 23 December 2014
Authors
Charles John Read
Department of Pure Mathematics
University of Leeds
Leeds LS2 9JT
United Kingdom