Chapter 7
Quasi diagonal
C* -Alge bras
In this chapter we present the basics of a large and intriguing class of C -
algebras. Quasidiagonality can be formulated in many ways, but the abun-
dance of characterizations and natural examples should not lead one to
believe that these algebras are easily understood. On the contrary, they can
be deceptively difficult.
Most of the material in this chapter is relatively easy, but there are
two important theorems which deserve recognition. The first is Voiculescu's
homotopy invariance theorem, Theorem 7.3.6, which shows (among other
things) that the natural K-theoretic operation of suspension always pro-
duces quasidiagonal C-algebras - i.e., for arbitrary A, SA= C 0 (0, 1)@ A
is quasidiagonal. The other important result, due to Dadarlat, is a strong
approximation theorem in the presence of exactness (Theorem 7. 5. 7).
7.1. The definition, easy examples and obstructions
As with nuclear and exact C*-algebras, we will develop the theory of qua-
sidiagonality in reverse chronological order. Our definition is in terms of
finite-dimensional approximations; the next section returns to the original
representation-theoretic notion.Definition 7 .1.1. A C* -algebra A is called quasidiagonal ( QD) if there ex-
ists a net of c.c.p. maps 'Pn: A --+ Mk(n) (C) which is both asymptotically
multiplicative (i.e., l\'Pn(ab) - 'Pn(a)cpn(b)\I--+ 0 for all a, b EA) and asymp-
totically isometric (i.e., \la\1 = limn->oo ll'Pn(a)i\ for all a EA).
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