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.
