1550075568-C-Algebras_and_Finite-Dimensional_Approximations__Brown_

70 3. Tensor Products

Proof. Since positive elements span, we may assume a 2: 0. In this case,

for each positive b E B we have that 7r( a Q9 b) is also a positive operator in

IIB(H) since

7r( a Q9 b) = 7r( al/2 Q9 b1;2)7r( al/2 Q9 bl/2) 2: O.

By the Closed Graph Theorem it suffices to show that if bn ----t 0 and

?r(a Q9 bn) ----t TE B(H) (in norm), then T = 0. Since every bounded linear

functional is a linear combination of positive functionals, it will follow that

T = 0 if we can show that r.p(T) = 0 for every positive functional cp.

So let cp be an arbitrary positive functional on IIB(H). Since 0:::; b =:::?-0:::;

7r( a Q9 b), it follows that b 1-+ cp o 7r( a Q9 b) defines a positive linear functional

on B and, as such, is necessarily bounded. Hence

cp(T) = lim cp( 7r( a Q9 bn)) = lim r.p o 7r( a Q9 bn) = 0

as desired. D

Theorem 3.2.6 (Restrictions). Let 7f: A0B ----t IIB(H) be a nondegenerate *-

representation. Then there exist nondegenerate *-representations 1fA: A ----t

IIB(H) and 1fB: B ----t IIB(H), with commuting ranges, such that

Proof. The question is not how to define the restrictions -there is only one
possibility-but rather, why the definition works. Since 7f is nondegenerate,
the vectors 1r(x)v, x E A 0 B and v E H, are dense in H. Hence we are
forced to define
7rA(a)(7r(x)v) = 7r(Laai Q9 bi)v
i
for a E A, x = I:i ai Q9 bi E A 0 B and v E H. We must show that this is
well-defined and bounded (hence it extends to a bounded operator on all of
H).
To prove this, we let {en} be an approximate unit for B and consider
the (well-defined, bounded) operators 7f (a Q9 en). For x = I:i ai Q9 bi E A 0 B
and v E H we have

117r(a@ en)(7r(x)v) - 7r(~ aai@ bi)vll = 117f ( ~ aai Q9 (enbi - bi)) vii

i i

:::; L Mi I/ enbi - bi I/ ----t 0,

i

where the Mi's are (a finite set of) constants depending on the elements aai
and coming from Lemma 3.2.5. Since there is a uniform bound on the norms
l/?r(a Q9 en)// (Lemma 3.2.5 again), it follows that our definition of 7rA(a) is
both well-defined and extends to a bounded operator on all H.

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