244 7. Quasidiagonal C* -Algebrasfor all T E ~ and
llPv-vll<c:
for all v EX·
As usual, this local formulation is handy when trying to verify quasidi-
agonality for a particular set and not so useful when proving something
about a known quasidiagonal set. To get a better characterization, we need
a lemma on perturbing projections.
Lemma 7.2.2. Let A be a unital C*-algebra and let p, q EA be projections.
(1) If llq -Pll < 1, then there is a unitary u E A with uqu* = p and
Ill -ull :S 4llP -qll ·
(2) If llq -pqll < 1/4, then there is a unitary element u EA such that
uqu* :::; p (with equality whenever llq - Pll < 1) and Ill - ull <
10llq-pqll-Proof. First assume liq -Pll < 1, consider x = pq and observe that
llxx -qll = llq(p -q)qll < l;
hence xx is invertible in qAq. Letting lxlq^1 E qAq be such that lxlq^1 lxl =
q and v = xlxlq1, we see that x = vlxl is the polar decomposition. In
particular, vv = q and Po = vv :::; p. But, xx is invertible in pAp
(xx - p = p(q - p)p) and so the range projection of x must be p - i.e.,
Po = vv* = p. We also have that
llq -vii :S llq - xii+ llvlxl - vii :S ll(q -p)qll + llv(lxl - q)ll :S 2llq -Pll
since 11 lxl - qll :::; llx*x - qll :::; llP - qll- Applying the same argument to
orthogonal projections we find a partial isometry w EA such that q1-= w*w,
p1-= ww* and llq1--wll :::; 2llq-pll· Defining our unitary u = v+w completes
the proof of the first assertion.
Now suppose that c: = llq-pqll < 1/4. If x = pq, we have llq-x*xll:::; c:;
hence lxl is invertible in qAq. As above, put v = xlxlq^1 and evidently
q = v*v and Po = vv* :::; p (with equality whenever llq -Pll < 1). Also, just
as above, llq -vii :::; llq - xii+ llvlxl - vii :::; 2c:, which implies llq -Poll :::; 4c:.
Now, since llq1--p~ll :::; 4c: < 1, the first assertion yields a partial isometry
w EA such that q1-= w*w, p~ = ww* and llq1--wll :::; 8c:. It follows that
again u = v + w is a unitary element with the right properties. D
Proposition 7.2.3. Let 0 c IIB(1i) be a norm separable, quasidiagonal set of
operators on a separable Hilbert space.^6 There exists an increasing sequence
of finite-rank projections P1 :::; P2 :::; · · · converging strongly to the identity
and such that II [Pn, T] II ---+ 0 for all TE n.(^6) We leave it to the reader to formulate and prove a nonseparable version.