Duality and operator algebras II: operator algebras as Banach algebras

We answer, by counterexample, several questions concerning algebras of operators on a Hilbert space. The answers add further weight to the thesis that, for many purposes, such algebras ought to be studied in the framework of operator spaces, as opposed to that of Banach spaces and Banach algebras. In particular, the 'nonselfadjoint analogue' of a w*-algebra resides naturally in the category of dual operator spaces, as opposed to dual Banach spaces. We also show that an automatic w*-continuity result in the preceding paper of the authors, is sharp.