Classification of Nuclear, Simple C*-algebras
Published by Springer Verlag
in the series
Encyclopaedia of Mathematical Sciences.
Publication: Fall, 2001.
Link to book: Hardback.
198pp. ISBN: 3-540-42305-X, price: EUR: 96,95
I list here the typographical errors, other corrections, and updates to the
book found since the proofs were submitted to the printers May
I shall gratefully receive - and post - any corrections the
readers have found in the book. Please address these to firstname.lastname@example.org.
List of corrections
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symbols, and the text has in parts been reworded to avoid math
symbols. You can also download the corrections written out
more legibly in TeX here:
Corrections: dvi file -
ps file - pdf file.
- p. 11, l. -12: It is not true in general that multiplier inner
automorphisms are approximately inner. However, it is true whenever A
is of stable rank one or wheneverA is stable. A counterexample to the
general statement is obtained eg. by taking A = C0(R
+,B), where B is the Bunce-Deddens algebra, and the
automorphism on A is conjungation by a unitary u in B
whose class in K1 is a generator.
- p. 11, Definition 1.1.15: In lieu of the comment above,
the unitary (unitaries) appearing in the definition of unitary
equivalence and approximate unitary equivalence of *-homomorphisms
should belong to the unitization of B (and not just to the multiplier
algebra of B).
- p. 26, l. 6: Replace ``or A admits an approximate
unit consisting of projection'' by ``or (A tensor K)
admits an approximate unit consisting of projection''.
- p. 32, Corollary 2.3.3 and its proof: The unitaries
un and vn should belong to the
unitization of Bn+1 and An,
respectively (rather than to their multiplier algebras).
- p. 33, l. 2: Replace "psik" with
- p. 87, l. -4: It is not true that the C*-algebras
C*(v1), C*(v2), ...,
C*(vr) commute with each other. Actually,
vi and vj anticommute. It is
nevertheless true that C*(v1, v2, ... ,
vr) is isomorphic to M2 tensor M2
tensor ... tensor M2 and that the isomorphism sigma
acts as claimed.
- p. 93, l. -10: Replace the reference "[84, Lemma 1.9]"
with "[84, Lemma 1.6]".
- p. 95, l. 14: Replace the reference "[84, Lemma 1.8]"
with "[84, Lemma 1.5]".
- p. 96, l. 5: Replace ``such that |xn| < epsilon''
with ``such that |xn - x0| < epsilon''.
- p. 97, Lemma 6.2.5: Assume in and above Lemma 6.2.5 that
B is a unital C*-algebra. In the proof of Lemma 6.2.5
replace Aomega by Bomega (p. 97,
l. -5), linfty(A) by linfty(B)
(p. 97, l. -4), A by B (p. 98, l. 2), and
linfty(A) by linfty(B) (p. 98,
- p. 100, l. 13: Replace "[84, Proposition 1.7]" with
"[84, Proposition 1.4]".
- p. 101, l. 13: Replace three occurances of
- p. 101, l. 17: Replace the reference "[84, Lemma 1.10]"
with "[84, Lemma 1.7]".
- p. 103, l. -11: Replace the second "=" in the display with
"less than or equal to".
- p. 105, Lemma 6.3.9: In the lemma and its proof we must replace
(O2)omega for any free ultrafilter omega. In the
proof of the lemma (eg. when one has to establish the inequalities on p. 106,
l. 5) it is important that the norm of piomega(x) is an actual
limit (along omega) rather than a limes superior.
Consequently, in the proof of Theorem 6.3.11 on page 108 we must
similarly change all occurances of (O2)infty with
(O2)omega. One must also modify the proof of
Lemma 6.3.11 on page 107.
- p. 105, l. -1: Replace "operator spaces" with "operator systems".
- p. 107, l. -8: The crossed product
C0(R,A) x Z is isomorphic to (K
tensor C(T) tensor A), not to (K
tensor A). It remains true that A embeds into (K
tensor C(T) tensor A), and hence into
C0(R,A) x Z.
Actually, one can obtain this
embedding in a nicer way: One observes that the crossed product
C0(R,A) x Z is isomorphic to
(C0(R) x Z) tensor A and that
C0(R) x Z contains a non-zero projection p.
One can then take the embedding to be: a maps to a tensor
p. The (Rieffel) projection p is of the form p = gu* + f +
ug, where u is the canonical unitary that implements the action
tau and f, g are (real valued) functions in
C0(R) given by f(t) = 1 - |t| when
-1 < t < 1, and 0 otherwise, and g(t) =
sqrt(t - t^2) when 0 < t < 1, and 0 otherwise.
- p. 109, l. 15: One must also show that the relative commutant
of A in Aomega is different from
C. This follows from the fact the the isometry
s (in this relative commutant) can be chosen to be
non-unitary. This, again,
follows from the fact that the isometry t in Lemma
6.3.2 actually is non-unitary (by construction). It follows
that the isometry s = ut from Proposition 6.3.3 likewise
is non-unitary. It finally follows from the proof of
Corollary 6.3.5 (ii) (second part) that the isometries
sn are non-unitary, finally making the
isometry s in the proof of Proposition 7.1.1
- p. 112-113, Proposition 7.2.5: The proof given of this
proposition only works when A is also assumed to be
nuclear. (The result by Phillips and Lin holds in the
stated generality.) More specifically,
in the proof of the "general case" on page 113
one needs nuclearity of
A to apply Proposition 7.1.1.
Fortunately, Proposition 7.2.5 is
applied to the nuclear C*-algebra A =
tensor Oinfty) in the proof of Theorem
- p. 124, Lemma 8.2.13: The map kappaA,B goes from
H^(A,B) to KK(A,B otimes Oinfty).
- p. 127, Theorem 8.3.3 (iii): The two unital *-homomorphisms
must also be injective. In part (b) homotopy must be replaced with
stable homotopy (i.e., homotopy relatively to B tensor the
- p. 132, l. -6: In the displayed formula in Corollary
8.4.11 (i), replace each occurence of
Question 8.4.4 on page 129 has been answered in the negative
in a revised version of .
Last modified: Thu Oct 19 16:44:46 CEST 2006