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
, the intersection of the
Jacobson radical of
with
is precisely
the radical of
(this is claimed for either of the Arens products on
). In
this paper we begin with a simple counterexample to that claim, in which
is a radical operator algebra, but not every element of
lies in the radical of
. We then develop a more
complicated example ,
which, once again, is a radical operator algebra, but
is semisimple.
So
is zero,
but
.
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
was
wrong; but we believe we are the first to provide “neat” counterexamples as
described.