B. Operator Spaces 457there exists a net {x.>.h of norm-one elements in Mn(X) which converges
toxin the cr(Mn(X),Mn(X))-topology. But since X C X completely
isometrically, by Lemma B.15, one has llx>-llMn(X) :::; 1 for all .. Since
each entry of X.>. converges in the weak-topology to the corresponding entry
of x, one has llxllMn(X) :::; 1. For the converse inequality, suppose this
time that llxllMn(X) :S 1. Then the corresponding map Tx: X ---+Mn((('.)
has llTxllcb :::; 1. It follows from Lemma B.12 that Tx can be approxi-
mated in the point-norm topology by weak-continuous complete contrac-
tions T.>. : X* ---+ Mn ( q. Then, the corresponding element x .\ E Mn ( X)
satisfies llx>-11 = llT>-llcb for every .. Since the net (x.>.).>. converges to
x in cr(Mn(X)*,Mn(X)), we have llxllMn(X) :S 1. This completes the
proof. D
If X is a dual operator space, then by definition there is a weak -
ultraweak homeomorphic complete isometry from X onto an ultraweakly
closed subspace of IB('li). On the other hand, if X c IB('li) is an ultraweakly
closed operator subspace, then the space X of ultraweakly continuous lin-
ear functionals on X is a predual of X, i.e., X = (X) canonically. Since
X c X as a Banach space and X is a (dual) operator space, we can equip
the predual X with an operator space structure via X c X. Fortunately,
the equality X = (X) holds completely isometrically as usual. (One should
verify this.) We note that even if an operator space X happens to have a
predual X as a Banach space, it does not mean X has an operator space
structure with X = (X) completely isometrically. This is because such X
need not have a weak*-ultraweak homeomorphic c.i. representation [115].
References. The theory of operator spaces took off after Ruan's axiomatic
formalization, although many notions had appeared earlier in papers of
Arveson, of Choi and Effros, etc. The notion of duality was introduced
by Blecher [21], Blecher and Paulsen [22] and Effros and Ruan [62]. The-
orems B.5 and B.7 are taken from [141], and Corollary B.11 is taken from
[59]. Standard text books for operator spaces and completely bounded maps
are [63, 152, 141].