3.6. Inclusions and The 'Irick 87We will soon introduce The Trick and provide the converse to the pre-
vious result, but first we consider two nice corollaries.
Corollary 3.6.3. If AC B, A is nuclear and C is arbitrary, then we have
a natural inclusion
A ®rnax C C B ®rnax C.Proof. Let n: A , IIB(H) be a representation and 'Pn: A , Mk(n) (C),
'lf;n: Mk(n) (C) , n(A) be c.c.p. maps converging to n in the point-norm
topology.^14 By Arveson's Extension Theorem we may assume that each of
the 'Pn's is actually defined on all of B. Letting : B , nA(A)" be any
point-ultraweak cluster point of the maps 'lj;n o t.pn: B _,AC nA(A)", we
get the c.c.p. extension of n required to invoke Proposition 3.6.2. D
Corollary 3.6.4. If A C B is a hereditary subalgebra, then for every C we
have a natural inclusion
A ®rnax C C B ®rnax C.
Proof. If {en} c A is an approximate unit, then the c.c.p. maps 'Pn: B ,
A, 'Pn(b) = enben have the property that 'Pn(a) , a for all a E A. With
this observation, the proof is similar to the previous corollary, so we leave
the details to the reader. D
Proposition 3.6.5 (The Trick). Let AC B and C be C-algebras, II· Ila
be a C -norm on B 0 C and II · ll,e be the C -norm on A 0 C obtained by
restricting II · Ila to A 0 CC B 0 C. If nA: A, IIB(H), nc: C , IIB(H) are
representations with commuting ranges and if the product -homomorphism
nA x no: A 0 C _, IIB(H)
is continuous with respect to II· ll,e, then there exists a c.c.p. map cp: B _,
nc(C)' which extends nA.
Proof. Assume first that A, B and C are all unital and, moreover, that
lA = lB. Let
nA x,e nc: A 0,e C _, IIB(H)
be the extension of the product map to A@,eC. Since A@,eC C B®aC, we
apply Arveson's Extension Theorem to get a u.c.p. extension : B ®a C _,
IIB(H). The desired map is just cp(b) = (b ®le).
To see that cp takes values in no( C)' is a simple multiplicative domain
argument. Indeed, ClB 0 C lives in the multiplicative domain of since
k1B®C = nc is a *-homomorphism. Since B®Clc commutes with ClE®C
14It's crucial that ?T: A-> ?T(A) be nuclear; the result need not hold if 7r(A) is replaced by
JIB(?-l).