On matricially normed spaces

Arveson and Wittstock have proved a "non-commutative Hahn-Banach Theorem" for completely hounded operator-valued maps on spaces of operators. In this paper it is shown that if T is a linear map from the dual of an operator space into a C*-algebra, then the usual operator norm of T coincides with the completely bounded norm. This is used to prove that the Arveson-Wittstock theorem does not generalize to "matricially normed spaces". An elementary proof of the Arveson-Wittstock result is presented. Finally a simple bimodule interpretation is given for the "Haagerup" and "matricial" tensor products of matricially normed spaces.