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: 
doi:10.1016/0022-1236(91)90100-J 
Copyright © 1991 Published by Elsevier Inc.
Completely bounded transformations of
-operator spaces
Received 18 May 1989.
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Abstract
If
is the commutant of a strictly cyclic unilateral weighted shift with a monotonically decreasing weight sequence, then we show that there is a natural isomorphism of the Banach space of bounded linear maps from
into
(
) with the Banach space of bounded linear maps of the trace class operators into
, where
is a separable, infinite dimensional Hilbert space. Under this isomorphism, an operator Φ from
into
(
) is completely bounded if and only if its image extends to a bounded map of the Hilbert-Schmidt operators into
. The proof shows that if Φ is only completely row bounded, then Φ is in fact completely bounded. The characterization of the completely bounded maps is then used to prove the existence of a family of completely unbounded representations of
into
(
).
