1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

B. Operator Spaces 455

where 1-i = EBmEN EBxEBallm(X) £~. Unless otherwise stated, we always as-

sume that the dual space X* is equipped with this operator space struc-

ture. We observe that the weak* -topology of X* agrees with the ultra-

weak topology of IIB(1-i) and in particular that if a net cp;. E Mn(X*) con-

verges to an element cp E Mn(X*) in the entrywise weak*-topology, then

ll'PllMn(X*) '.S liminf;. ll'P>.llMn(X*)·

For cp = [cpi,j] E Mn(X*), we define Tep: X 3 x 1---+ [cpi,j(x)] E Mn(<C). It

follows from the definitions that

ll'PllMn(X*) = sup{ll(Bx)n(cp)llMn(Mm(<C)): m EN, XE Ballm(X)}

=sup{ II (Tep)m(x)llMm(Mn(<C)) : m EN, XE Ballm(X)}

= II Tep llcb,

where we used the standard identification Mn (Mm ( <C)) = Mm (Mn ( <C)). This
implies that the bijection

is isometric. Hence, we sometimes identify cp with Tep (and Mn(X*) with
CB(X,Mn(<C))). This identification can be generalized as follows.

Theorem B.13. Let X and Y be operator spaces. For z = L:k 'Pk® Yk E
X* 0 Y, we define a finite-rank map Tz: X---+ Y by

Tz(x) = L 'Pk(X)Yk·

k

This correspondence is isometric, meaning llzllx•@Y = llTzllcb, and hence

extends to a canonical (isometric) inclusion

X* ® Y c CB(X, Y).

Moreover, the inclusion is bijective if either X or Y is finite-dimensional.

Proof. Let Y c IIB(1-i). For simplicity, we assume that 1-i is separable and let
P 1 :::::; P 2 :::::; · · · be an increasing sequence of finite-rank projections in IIB(1-i)
with rank(Pn) = n and Pn /' 1 in the strong operator topology. We denote
by n: IIB(1-i) ---+ IIB(Pn1-i) ~ Mn(<C) the compression by Pn. Fix z EX* 0 Y.
Since T(id®<Pn)(z) = n o Tz in CB(X, Mn(<C)), we have

llzllmin =sup ll(id ® <I>n)(z)llx*®Mn(<C)

n

=sup ll<I>n ° Tzllcb

n