Abstract
We give a simple criterion for certain Banach algebras to be Arens regular, which applies in particular to the algebras ${l^1}$ with pointwise multiplication, ${L^\infty }(G)$, where $G$ is a compact group with convolution, and the trace-class algebra. This criterion is best established in the more general context of the regularity of bilinear maps, and depends on the existence of extensions of such maps.
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