78 3. Tensor ProductsThe next two results will allow us to deduce that certain states on tensor
products can always be decomposed into a product of states.
Lemma 3.4.2. If A C C, B = A' n C is the relative commutant of A and
if e is a state on c such that elA is a pure state on A, then e(ab) = e(a)e(b)
for all a E A, b E B.Proof. By a standard reduction we may assume that b E B is a positive
element of norm one and we will show e(ab) = e(a)e(b) for all a EA.
Case 1: Assume e(b) = 0. Then for all a EA we have that
le(ab)J^2 = 1e((ab^1 l^2 )(b^1 l^2 ))J^2 ~ e(aba*)e(b),
by the Cauchy-Schwarz inequality. Hence e(ab) = O = e(a)e(b).
Case 2: Assume e(b) = 1. Then (passing to unitizations, if necessary)
e(l - b) = 0, so by Case 1 we have 0 = e(a(l - b)) = e(a) - e(ab). Thus
e(ab) = e(a) = e(a)e(b).
Case 3: Assume 0 < e(b) < 1. Then we havee (a) = e ( b) ( e tb) e (ab)) + ( 1 - e ( b)) ( 1 -le ( b) e (a ( 1 - b))).
This equation shows that the restriction of e to A is a convex combination
of states on A. (Since b commutes with A, the functionals a f--+ efb) e( ab)
and a f--+ l-~(b)e(a(l - b)) are states.) But we assumed this restriction to
be pure and so it follows that e(a) = db)e(ab) for all a EA, so the proof is
complete. DLet JJ ·Ila be a C*-norm on A 8 B, A ®a B be the completion and e be
a state on A ®a B. We define the restrictions elA and elB as follows. Let
(7re, He, ve) be the GNS triplet and 1fe,A and 7re,B be the restriction homo-
morphisms given by Theorem 3.2.6. Then define elA(a) = (1re,A(a)ve, ve)
and elB(b) = (1re,B(b)ve,ve).
Corollary 3.4.3. Let II. Ila be a C* -norm on A 8 B and e be a state on
A®aB.
(1) If the restriction e1A is pure, then elA0B = eJA 8 elB·
(2) If A is abelian and e is a pure state on A ®a B, then both elA and
eJB are pure and elA8B = elA 8 elB·Proof. Assume that eJA is pure. Applying the previous lemma to the vector
state T f--+ (Tve, ve) on IIB(He), it follows that
~(a® b) = (1re,A(a)1fe,B(b)ve, ve)
= (1re,A(a)ve, ve)(1re,B(b)ve, ve)