Friday, April 4, 2014 — 3:30 PM EDT

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.

Location 
MC - Mathematics & Computer Building
5136B
200 University Avenue West

Waterloo, ON N2L 3G1
Canada

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