1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

102 3. Tensor Products

Proof. ( =?-) This is very similar to the proof of Lemma 3.6.10. If 'Pn: A---+

Mk(n)(C) and '!/Jn: Mk(n)(C)---+ Mare c.c.p. maps whose compositions con-

verge point-ultraweakly to cp, then we can define a sequence of c.c.p. maps

cl>n: A@M'---+ IIB(1i) by cl>n =('!/Jn Xmax lM1) o (cpn ® lM1). In other words,

we take the composition

A@ M' 'P~M' Mk(n)(C)@ M^1 1/Jn';<,!!!:!:!;/M' JIB(Ji),

where the maximal product map on the right exists thanks to Exercise 3.5.1
(and uniqueness of the C-norm on Mk(n)(C) 8 M'). Letting cl>: A@ M'---+
IIB(1i) be a point-ultraweak cluster point of the cl>n's, we get a c.c.p. map
which evidently extends the product map cp x lM': A 8 M' ---+ IIB(1i); this
direction of the proof is comple-te.
( ~) Now assume that cp x LM': A 8 M' ---+ IIB(1i) is min-continuous.
According to Proposition 3.8.2 we only have to show that cp belongs to the
point-ultraweak closure of the factorable maps. Thus we fix a finite set
i C A, a finite set of normal states S c M and E > 0. Let f E M* be
the average of all the states in S and note that f dominates a multiple of
each e E S. Thus, by the Radon-Nikodym-type result above, we can find
elements Yt, E 7r(M)' such that

e(m) = (7r(m)yt,lM, lM),

for all m E M and all e E S, where ?T: M ---+ IIB(L^2 (M, f)) is the GNS

representation.

Since Lemma 3.8.4 tells us that the product map

(?Tocp) x l7r(M)': A8?T(M)'---+IIB(L^2 (M,f))

is also min-continuous, we can define a stateμ on A@?T(M)' by the formula

μ(a® y) = (7r(cp(a))ylM, iM)·

Note that e(cp(a)) =μ(a® Yt,) for all a EA and e ES.
Let A® 1r(M)' C IIB(JC ® L^2 (M, f)) be a representation containing no
nonzero compacts (i.e., inflate any faithful representation of A to ensure Ac
IIB(JC) contains no compacts). By Glimm's lemma (Lemma 1.4.11) we can
approximate the functionalμ by vector states on IIB(JC@L^2 (M, f)). In partic-
ular, we can find an orthonormal set {v1, ... , vn} c JC and {b 1 , ... , bn} c M
such that
n n
lμ(a ® Yt,) - (a® Yt,(Lvi ®bi), L Vi® bi)I < E
i=l i=l

for all a E i and all e E S.