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 list here the typographical errors, other corrections, and updates to the book found since the proofs were submitted to the printers May 2001.

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

The list is displayed below in a mixture of html and TeX 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.

__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 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}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*and_{n}*v*should belong to the unitization of_{n}*B*and_{n+1}*A*, respectively (rather than to their multiplier algebras)._{n}**p. 33, l. 2:**Replace "psi_{k}" with "psi_{k+1}"**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*and_{i}*v*anticommute. It is nevertheless true that C*(_{j}*v*) is isomorphic to M_{1}, v_{2}, ... , v_{r}_{2}tensor M_{2}tensor ... tensor M_{2}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*| < epsilon'' with ``such that |_{n}*x*| < epsilon''._{n}- x_{0}**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 (*O*_{2})_{infty}with (*O*_{2})_{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 (*O*_{2})_{infty}with (*O*_{2})_{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 C_{0}(**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 C_{0}(**R**,*A*) x**Z**.

Actually, one can obtain this embedding in a nicer way: One observes that the crossed product C_{0}(**R**,*A*) x**Z**is isomorphic to (C_{0}(**R**) x**Z**) tensor*A*and that C_{0}(**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 C_{0}(**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*are non-unitary, finally making the isometry_{n}*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*tensor_{infty}*O*) in the proof of Theorem 7.2.6._{infty}**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*by_{nj}*M*(_{kj}*O*)_{nj}

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

Mikael Rordam Last modified: Thu Oct 19 16:44:46 CEST 2006