18 1. Fundamental Factseasily seen from explicit form of the correspondence). Unfortunately the cp .A's
need not be contractive, but they are "almost contractive" (since 'P.>-(1) -+
cp(l) in norm); hence we can fiddle with their norms a bit to correct this
deficiency. D1.7. Voiculescu's Theorem
Voiculescu's Theorem is analogous to the Hahn-Banach Theorem in two
ways: It gets used all of the time; and it really refers to a collection of
related results and corollaries.^4 Here, we collect all the forms we need,
though we only prove those which haven't yet appeared in a book.Finite-dimensional case. Exploiting the duality between c.p. maps A-+
Mn(C) and states on Mn(A), it is not too hard to deduce the next result
from Glimm's lemma (Lemma 1.4.11).
Proposition 1. 7.1. Let H be separable, 1 E A c JE(H) be a separable C*-
algebra and cp: A-+ Mn(C) be a u.c.p. map such that 'PIAnlK(7i) = 0. Then
there exist isometries Vk : .e;_ -+ H with the following properties:
(1) the ranges of the Vk 's are pairwise orthogonal;
(2) jjcp(a) - Vk*aVkll -+ 0 for every a EA.General case.
Definition 1. 7.2. Two maps 7r: A -+ JE(H) and O": A -+ JE(JC) are called
approximately unitarily equivalent if there is a sequence of unitary operators
Un: H-+ JC such that
llO"(a) - Un7r(a)U~ll-+ 0
for all a E A. If it also happens that O"( a) -Un7r( a) U~ is a compact operator,
for all a E A and n E N, then we say that 7r and O" are approximately unitarily
equivalent relative to the compacts.Note that approximate unitary equivalence relative to the compacts is a
much stronger notion as it implies that after passing to the Calkin algebra,
the representations 7r and O" are actually unitarily equivalent. See [11] or
[53, Corollary II.5.5] for a proof of the next result.
Theorem 1.7.3 (Voiculescu's Theorem). Let H and JC be separable Hilbert
spaces and A C JE(H) be a separable C*-algebra such that l'li E A. Let
1,: A ~ JE(H) denote the canonical inclusion and let p: A -+ JE(JC) be any
unital representation such that PIAnlK('li) = 0. Then 1, and 1, E9 p are approxi-
mately unitarily equivalent relative to the compacts.
(^4) Thirdly, some authors assume familiarity with all possible formulations and don't bother
explaining which version is being invoked.