248 7. Quasidiagonal C* -Algebras
Given a EA, of norm one, we consider the inner derivation
Da(b) = ba - ab, b EA.
Assuming b also comes from the unit ball, one uses induction on the inequal-
ity
llDa(bi+l)ll = llbDa(bi) + Da(b)bill :S llDa(bi)ll + llDa(b)ll
to deduce that
for all natural numbers j.
So, letllDa(bi)ll :S jllDa(b)llE
15 = n.
I:i=l iltil
and assume that 11 D a ( e) 11 = 11 [ e, a J 11 < 15 for some positive element e in the
unit ball of A. Then we have
n
II [f (e), a] II = llt1Da(e) + · · · + tnDa(en) II :S L ltil(illDa(e) II) < E
i=l
as desired. D
The notions of homotopic maps and homotopy equivalence of spaces are
fundamental in topology. Here is how these ideas translate into C*-lingo.
Definition 7.3.3. Let ao: A ----+ B and a1: A ----+ B be -homomorphisms.
Then we say ao and a1 are homotopic if there exist -homomorphisms
(ft : A ----+ B, 0 :S t :S 1, such that &o = ao, &1 = a1 and for every fixed
a EA the map [O, 1]----+ B, ti--+ 0-t(a), is continuous from the usual topology
on [O, 1] to the norm topology on B - in other words, there should exist -
homomorphisms which give norm continuous paths from ao(a) to a1(a), for
every a E A. If you prefer, this is equivalent to asking for a -homomorphism
A----+ C[O, l] 0 B which is ao at the left endpoint and a 1 at the right.
Definition 7.3.4. A C-algebra A is said to homotopically dominate an-
other C -algebra B if there are -homomorphisms 7r: B ----+A and a: A ----+ B
such that a o 7r is homotopic to idB. The algebras A and B are homotopy
equivalent if there exist -homomorphisms 7r : B ----+ A and a : A ----+ B such
that a o 7r is homotopic to idB and 7r o a is homotopic to idA.
We will show that if two C*-algebras are homotopy equivalent and one is
QD, then so is the other. The proof is much easier if we isolate a preliminary
step.
Proposition 7.3.5. Let ao, a1: B ----+ C be homotopic *-homomorphisms
such that ao is injective and a 1 (B) is a QD subalgebra of C. Then Bis QD.