Charles Read, University of Leeds
"Radicals of operator algebras: a counterexample to a well-known theorem"
A
paper
of
Gulick
from
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
$A$.
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
$\text{rad}(A^{**})
\cap
A
=
\{0\}$,
but
$\text{rad}(A)
=
A$.
We
examined
at
least
8
subsequent
papers
that
refer
to
Gulick's
paper,
and
we
find
that
most
authors
have
used
the
correct
material
and
have
avoided
using
the
wrong
result.
We
reckon,
then,
that
we
are
not
the
first
to
suspect
that
the
result
$\text{rad}(A^{**})
\cap
A
=
\text{rad}(A)$
was
wrong;
but
we
believe
we
are
the
first
to
provide
"neat''
counterexamples.