Classification of Nuclear, Simple C*-algebras

By Mikael Rørdam.

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 shall gratefully receive - and post - any corrections the readers have found in the book. Please address these to mikael@imada.sdu.dk.

List of corrections

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Corrections:

• 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 u_n and v_n should belong to the unitization of B_n+1 and A_n, respectively (rather than to their multiplier algebras).

• p. 33, l. 2: Replace "psi_k" with "psi_k+1"

• p. 87, l. -4: It is not true that the C*-algebras C*(v₁), C*(v₂), ..., C*(v_r) commute with each other. Actually, v_i and v_j anticommute. It is nevertheless true that C*(v₁, v₂, ... , v_r) 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 |x<sub>n</sub>| < epsilon'' with ``such that |x_n - 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 A_omega by B_omega (p. 97, l. -5), l^infty(A) by l^infty(B) (p. 97, l. -4), A by B (p. 98, l. 2), and l^infty(A) by l^infty(B) (p. 98, l. 3).

• p. 100, l. 13: Replace "[84, Proposition 1.7]" with "[84, Proposition 1.4]".

• p. 101, l. 13: Replace three occurances of phi(1_A) with phi(1_Mn).

• 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)_infty with (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 pi_omega(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 A_omega 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 s_n are non-unitary, finally making the isometry s in the proof of Proposition 7.1.1 non-unitary.

• 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 = (O_infty tensor O_infty) in the proof of Theorem 7.2.6.

• p. 124, Lemma 8.2.13: The map kappa_A,B goes from H^(A,B) to KK(A,B otimes O_infty).

• 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 compacts.

• p. 132, l. -6: In the displayed formula in Corollary 8.4.11 (i), replace each occurence of O_nj by M_kj(O_nj)

Question 8.4.4 on page 129 has been answered in the negative in a revised version of [122].