1.6. Arveson's Extension Theorem 17Corollary 1.5.16. Let EC A be an operator subsystem and cp: E ----t Mn(C)
be a c.p. map. Then cp extends to a c.p. map A ----t Mn(<C).Proof. Given cp, we can define a linear functional r:p on Mn(E) (as above)
and it is positive (same proof as above). By the previous lemma, we can
extend to a positive functional on all of Mn(A) and then apply the one-to-one
correspondence in reverse to get our desired extension. D1.6. Arveson's Extension Theorem
The following theorem, due to Arveson, is absolutely fundamental and prob-
ably gets invoked more than any other result in these notes.Theorem 1.6.1. Let A be a unital C*-algebra and E C A be an operator
subsystem. Then, every c.c.p. map cp: E ----t IIB(H) extends to q c.c.p. map
<p: A----t IIB(1i).
Proof. Let Pi E IIB(H) be an increasing net of finite-rank projections which
converge to the identity in the strong operator topology. For each i, we
regard the c.c. p. map 'Pi: E ----t PiIIB(1i)~, 'Pi ( e) = ~cp( e )Pi as taking values
in a matrix algebra. Thus, by Corollary 1.5.16, we may assume that each 'Pi
is actually defined on all of A. Now we regard 'Pi as taking values in IIB(H)
and apply compactness of the unit ball of IIB(A, IIB(H)) in the point-ultraweak
topology (Theorem 1.3. 7) to find a cluster point : A ----t IIB(H). It is readily
verified that is c.p. and extends cp. D
Remark 1.6.2 (Injectivity and Arveson's Theorem). Arveson's Extension
Theorem is equivalent to the statement that IIB(1i) is injective in the category
of operator systems with c.c.p. maps as morphisms. This is even true in the
category of operator spaces with completely bounded maps as morphisms.
It follows from Arveson's Theorem that a von Neumann algebra M c
IIB(1i) is injective if and only if there is a conditional expectation from IIB(H)
onto M. It also follows that injectivity is independent of the choice of faithful
representation MC IIB(H).
Corollary 1.6.3. Let E c IIB(H) be an ultraweakly closed operator system
and let cp: E ----t Mn(C) be a c.c.p. map. There exists a net of ultraweakly
continuous c.c.p. maps cp;..: E ----t Mn(C) which converges to cp in the point-
norm topology (i.e., llcp;..(x) - cp(x)ll ----t 0 for all x EE).Proof. By Arveson's Extension Theorem, we may assume that E = IIB(H).
Since (p is c.p., the corresponding functional cp E Mn(IIB(H) )* is positive.
Hence, there exists a net cp;.. of positive normal linear functionals which
converges pointwise to cp. Then, the corresponding c.p. maps cp;..: IIB(H) ----t
Mn(C) are normal and converge to cp in the point-norm topology (which is