1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

3.9. Exactness and tensor products 109

Proposition 3.9.6. Let A C IIB(H) be a unital ®-exact C* -algebra, { Pn}

be any increasing sequence of finite-rank projections converging strongly to

17-l and E C A be any finite-dimensional operator system. If !.pn: E --+

Ms(n)(<C) ~ PnIIB(H)Pn are defined by !.pn(x) = PnxPn, then we have

llcp;;:-^1 1cpn(E) llcb--+ 1.^28

Proof. Just to ease notation, let us assume that the rank of Pn is n so that
PnIIB(H)Pn ~ Mn(<C). Modulo notational complications, the same proof
works in general.

Proceeding by contradiction, we'll show that if limn-+oo llcp~^1 lcpn(E) llcb =
f3 > 1, then it is possible to construct an element

ITn(E ® Mk(n)(<C))

XE ffin(E ® Mk(n)(<C))'

for a suitable choice of k(n)'s, such that llXll = 1 _but under the mapping

ITn (E ® Mk(n) (<C)) --+ E ® ( ITn Mk(n) (<C))

ffin (E ® Mk(n) (<C)) EBn Mk(n) (<C)

X goes to an element of norm:::; 13-^1 < 1. Since A is ®-exact, Lemma 3.9.5
will give our contradiction.

Thus we assume that limn-+oo llcp~^1 1cpn(E)llcb = /3 > 1. Since the maps
cp~^1 1cpn(E) are expanding some elements (after tensoring with large enough
matrices), it follows that the original maps !.pn have to be shrinking some el-
ements (after tensoring with large enough matrices). In other words, we can
find a sequence of integers k(n) and an element (Xn)n E I1n(E ®Mk(n)(<C))
such that llXnll = 1 for all n and

Denote by

n-+oo lim IJcpn®idk(n)(Xn)ll =/3-^1 <1.

XE I1n(E ® Mk(n)(<C))

ffin(E ® Mk(n)(C))

the canonical image of (Xn)n E ITn(E ® Mk(n)(<C)).

Now apply the contractive map (Lemma 3.9.5)

ITn (E ® Mk(n) (<C)) --+ E ® ( ITn Mk(n) (<C))

EBn (E ® Mk(n) (<C)) EBn Mk(n) (<C)

28Since Pn :<; Pn+l, we have 'Pn = cpn o 'Pn+l = 'Pn+l o 'Pn· Hence if 'Pn+l compresses the

norm of some operator, then 'Pn can only shrink it further; that is, 1 :<; JJcp;;:i 1 J'Pn+i(E)llcb :<;

llcp;;:^1 J'Pn(E) llcb for all n. In particular, this shows lim llcp;;:^1 Jcpn(E) JJcb exists.