Corrections to “Classification of Nuclear, Simple C^{∗}-algebras”

• 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 whenever A is stable. A counterexample to the general statement is obtained eg. by taking A = C_{0}(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 K_{1}(B) 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 B̃ of B (and not just to the multiplier algebra M(B) of B).

• p. 26, l. 6: Replace “or A admits an approximate unit consisting of projection” by “or A ⊗ 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 “φ_{k+1} ∘ ψ_{k} ∘ φ_{k}” with “φ_{k+1} ∘ ψ_{k+1} ∘ φ_{k}”.

• p. 87, l. -4: It is not true that the C^{∗}-algebras C^{∗}(v_{1}), C^{∗}(v_{2}), …, C^{∗}(v_{r}) commute with each other. Actually, v_{i} and v_{j} anticommute. It is nevertheless true that C^{∗}(v_{1}, v_{2}, ⋯, v_{r}) is isomorphic to M_{2} ⊗ M_{2} ⊗ ⋯ ⊗ M_{2} and that the isomorphism σ 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_{n}| < ε” with “such that |x_{n} − x_{0}| < ε”.

• 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_{ω} by B_{ω} (p. 97, l. -5), ℓ^{∞}(A) by ℓ^{∞}(B) (p. 97, l. -4), A by B (p. 98, l. 2), and ℓ^{∞}(A) by ℓ^{∞}(B) (p. 98, l. 3).

• p. 100, l. 13: Replace the reference “[84, Proposition 1.7]” with “[84, Proposition 1.4]”.

• p. 101, l. 13: Replace three occurances of “φ(1_{A})” with “φ(1_{M_{n}(ℂ)})”.

• p. 101, l. 17: Replace the reference ”[84, Lemma 1.10]” with ”[84, Lemma 1.7]”.

• p. 103, l. -11: The second “=” in the display should be replaced with “≤”.

• p. 105, Lemma 6.3.9: In the lemma and its proof we must replace (O_{2})_{∞} with (O_{2})_{ω} for some free ultrafilter ω. 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 π_{ω}(x), where x is an element in ℓ^{∞}(O_{2}), is an actual limit: lim_{ω} ∥x_{n}∥, rather than lim sup_{n→∞} ∥x_{n}∥.

Consequently, in the proof of Theorem 6.3.11 on page 108 we must similarly change all occurances of (O_{2})_{∞} with (O_{2})_{ω}. One must also modify the proof of Lemma 6.3.10: Using standard facts about quasidiagonal C-algebras, one can show that the embedding φ can be chosen such the composition of the embedding ῑ ∘ φ with the natural surjection (O_{2})_{∞} → (O_{2})_{ω} yields an embedding A → (O_{2})_{ω}. Now the (revised) Lemma 6.3.9 applies

• p. 105, l. -1: Replace “operator spaces” with “operator systems”.

• p. 107, l. -8: The crossed product C_{0}(ℝ, A) ⋊_{τ} ℤ is isomorphic to K ⊗ C(𝕋) ⊗ A, not to K ⊗ A. It remains true that A embeds into K ⊗ C(𝕋) ⊗ A and hence into C_{0}(ℝ, A) ⋊_{τ} ℤ.

Actually, one can obtain this embedding in a nicer way: One observes that the crossed product C_{0}(ℝ, A)⋊_{τ} ℤ is isomorphic to (C_{0}(ℝ)⋊_{τ} ℤ)⊗A, and that C_{0}(ℝ)⋊_{τ} ℤ contains a non-zero projection p. One can then take the embedding to be: a ↦ a ⊗ p.

The (Rieffel) projection p is given by p = gu^{∗} + f + ug, where u (in the multiplier algebra of C_{0}(ℝ) ⋊_{τ} ℤ) is the canonical unitary that implements the action τ and f, g ∈ C_{0}(ℝ) are given by

f(t) = \begin{cases}1 − |t|, & −1 ≤ t ≤ 1, \\ 0, & otherwise, \end{cases} g(t) = \begin{cases}\sqrt{t − t^{2}}, & 0 ≤ t ≤ 1, \\ 0, & otherwise. \end{cases}

• p. 109, l 15: One must also show that the relative commutant A_{ω} ∩ A′ is different from ℂ. This follows from the fact the the isometry s ∈ A_{ω} ∩ A′ 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_{∞} ⊗ O_{∞} in the proof of Theorem 7.2.6.

• p. 124, Lemma 8.2.13: Replace “The map κ_{A,B}: KK(A, B) → Ĥ(A, B)..” with “The map κ_{A,B}: Ĥ(A, B) → KK(A, B ⊗ O_{∞}))..”

• p. 127, Theorem 8.3.3 (iii): The two unital, nuclear ^{∗}-homomorphisms φ and ψ must also be injective. In (b), one must replace φ ∼_{h} ψ with stable homotopy, i.e., that φ and ψ are homotopic as maps A → B ⊗ K (viewing B as a sub-algebra of B ⊗ K).

• p. 132, l. -6: Replace the last two lines of Corollary 8.4.11 (i) by:

(M_{k_{1}}(O_{n_{1}}) ⊕ M_{k_{2}}(O_{n_{2}}) ⊕ ⋯ ⊕ M_{k_{r}}(O_{n_{r}})) ⊗ C(𝕋),

for some r ∈ ℕ, some k_{1}, k_{2}, …, k_{r} ∈ ℕ, and some n_{1}, n_{2}, …, n_{r} in {2, 3, … }.

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