Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 447-451.

The flip is often discontinuous

Authors: Volker Runde
Author institution: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Summary: Let $\mathfrak A$ be a Banach algebra. The flip on $\mathfrak A \otimes \mathfrak A^\rm {op}$ is defined through $\mathfrak A \otimes \mathfrak A^\rm {op} \ni a \otimes b \mapsto b \otimes a$. If $\mathfrak A$ is ultraprime, $\mathcal{El}(\mathfrak A)$, the algebra of all eleme ntary operators on $\mathfrak A$, can be algebraically identified with $\mathfrak A \otimes \mathfrak A^\rm {op}$, so that the flip is well defi ned on $\mathcal{El}(\mathfrak A)$. We show that the flip on $\mathcal{El}(\mathfrak A)$ is discontinuous if $\mathfrak A = {\cal K}(E)$ for a reflexive Ban ach space $E$ with the approximation property.

Keywords: Elementary operators, flip, Arens products

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