456 B. Operator SpacesRemark B.14 (Technical observation). Let Ebe a finite-dimensional op-
erator space and A be a C -algebra. Assume we are given elements z E.
E Q9 A and Zi E E* Q9 A. Let Tz : E ---+ A* and Tzi : E ---+ A be the corre-
sponding c.b. maps. Note that there is an algebraic (though not necessarily
isometric) linear isomorphism E Q9 A ~ (E* Q9 A) (Proposition 9.2.1).
We claim that if Zi---+ z in the weak-topology coming from (E @A), then
Tzi ---+ Tz in the point-ultraweak topology.
The point is that finite-dimensionality of E gives an algebraic identifi-
cation (E* Q9 A)*= E Q9 A*. Hence, any~ E (E* Q9 A)* can be written~ = I>j Q9 T]j,
where ej E E and T]j E A*. Straightforward calculations, using nothing but
the definitions, show convergence works as asserted.
Suppose that X and Y are operator spaces and <p: X ---+ Y is a finite-
rank weak-continuous c.b. map. By the above theorem, <p corresponds to
an element z E X0Y (rather than z E X0Y) with ll'Pllcb = llzllx@Y· It
is natural to ask whether llzllx@Y = llzllx@Y for z EX 0 Y. Fortunately,
this is true as the canonical inclusion of X into X is completely isometric
for any operator space X.
Lemma B.15. For any operator space X, the canonical inclusion X c X**
is completely isometric.
Proof. For x = [xi,j] E Mn(X), we have
llxllMn(X) = sup{ll'Pn(x)llMn(Mm(IC)): m EN, <p: X ---t Mm(CC), ll'Pllcb ::=; 1}
= sup{ll(Bx)m('P)llMm(Mn(C)) : m EN, <p E Mm(X*), ll'Pll·:s; 1}
= llBxllcB(X*,Mn(C)) ~ llxllMn(X**)since CB ( X*, Mn ( tC)) = Mn ( X**) by definition. D
Let X be an operator space. There is a canonical algebraic identification
Mn ( X) = Mn ( X) . As one might expect, under this identification a net in
Mn(X) converges in the (J(Mn(X), Mn(X)*)-topology if and only if each
matrix entry converges in the (J(X*, X)-topology. Moreover, Mn(X) =
Mn(X) is a completely isometric identification.
Proposition B.16. Let X be an operator space. The canonical isomor-
phism Mn(X) ~ Mn(X) is (completely) isometric. In particular, if A
is a C* -algebra, then the operator space structure of A** as a second dual
operator space agrees with the von Neumann algebra structure.
Proof. We will show llxllMn(X) = llxllMn(X) for every x = [xi,j] E
Mn(X). Suppose first that llxllMn(X) :::; 1. By the bi-polar theorem,