An Introduction to K-theory for C*-algebras

By Mikael Rørdam, Flemming Larsen, and Niels Jakob Laustsen.

Published by Cambridge University Press in the series London Mathematical Society Student Texts. Publication date: July 20th, 2000. Links to book: Paperback, Hardback.

Bibliographic information: 228 x 152 mm 256pp.

Hardback: ISBN: 0 521 783348, price: £ 60.00. Paperback: ISBN: 0 521 789443, price: £ 20.99.

We shall gratefully receive - and post - any corrections our readers have found in the book. Please address these to Mikael Rørdam at rordam@math.ku.dk.

List of corrections

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 - corrections: pdf file.

For the convenience of the reader, the corrections are divided into two categories depending on the urgency that they be implemented.

Necessary corrections:

• Page 5, l. 5: Replace "one and only one *-homomorphism ..." by "one and only one unital *-homomorphism ...".

• Page 7, item (ii): Replace "X is compact" by "X is compact and metrizable".

• Page 7, item (iv): This statement has to be modified when X and Y are not compact and when phi is not unital. In the first statement, one must insert the word "proper" in front of "continuous function" two places. (A continuous function f : Y -> X is said to be proper if for every compact subset K of X, its preimage, f^-1(K), is a compact subset of Y.) In the second statement one must require that the image of phi is not a proper ideal of C₀(Y).

• Page 11, l. -9: Replace "for all x, y in A, and..." with "for all x, y in A^~, and...".

• Page 14, Exercise 1.14: Replace "if and only if" by "if".

• Page 19, l. 12: Replace "retract" by "deformation retract".

• Page 20, l. 4: Replace "each bounded subset Omega of A" by "each bounded subset Omega of A⁺".

• Page 41, l. -2: Replace "a collection of maps phi -> F(phi)" by "a map phi -> F(phi)"

• Page 46, 3.3.1: One should assume that the C*-algebra A is unital; at least in the last section, where K₀(tau) is defined.

• Page 53, l. 9: Change "Example 3.3.5" to "Example 3.3.4".

• Page 55, Exercise 3.4 (iii): delete "over C(X)" from "rectangular matrices v₁, v₂,..., v_r over C(X) such that..."

• Page 74, l. 15: Replace ``Use (iii) to show...'' with ``Use (iv) to show...''.

• Page 83, l. 14: After "for each pair of commuting elements a,b in A⁺" add ", and such that tau extends (possibly in a non-canonical way) to a continuous function tau : M₂(A)⁺ -> R⁺ with the same properties."

• Page 84, l. 6-7: Replace "every unital, stably finite, separable, exact C*-algebra admits a faithful trace." by "every unital, stably finite, exact C*-algebra admits a tracial state.".

• Page 94, Proposition 6.2.4 (iii): the right-hand side of the equation should read: {a in A_n : lim_{m -> infty} ||phi_m,n(a)|| = 0}.

• Page 100, l. -4: Replace "By Proposition 6.4.2 (iii)" by "By Proposition 6.4.2 (ii)".

• Page 102, l. -2: Replace "K₀(A_n) -> ..." with "K₀(A) -> ...".

• Page 103, l. 4: Replace "... = K₀(g') = g," with "... = K₀(kappa_n)(g') = g,".

• Page 103, l. 12-13: Replace "[31, Section 3.3]" with "[31, Section 3.4]".

• Page 152, Exercise 8.18 (iv): replace "a" by "c" in the second sentence "Show that there is an invertible element b in A with [b]₁ = [a]₁ in K₁(A)."

• Page 155, l. 11: Replace "and p in U _2(n1+n2)(I^~)" by "and p in P _2(n1+n2)(I^~)".

• Page 167, l. -11: Replace "an isomorphism" with "injective" in "Moreover, phi is an isomorphism if and only if v..."

• Page 169, Eq. (9.13): Move the minus sign from the lower left corner of the matrix to the upper right corner.

• Page 188, l. 14: The inclusion between the two sets GL_n((SA)^~) and U_n((SA)^~) must be reversed.

• Page 188, l. 15: Replace "U_n((SA)^~)" with "GL_n((SA)^~)".

• Page 195, l. -9: Replace two occurances of "1_A" with "1_B".

• Page 196, l. 9-10: The inclusion between the two sets P_n(A) and GI_n(A) must be reversed.

• Page 203, l. 6: Replace "retract" by "deformation retract".

• Page 203, l. 14-15: It is not true that the two sets {u in C([0,1],V(A)) : u(0)=u(1)=1} and {u in U_infty(SA) : s(u)=1} are equal, but but {u in U_infty(SA) : s(u)=1} is a dense subset of {u in C([0,1],V(A)) : u(0)=u(1)=1}, and this inclusion is a pi_0-equivalence (which justifies the claim in line 17). (Use density and the fact that if u,v are elements of either set and if ||u-v|| < 1, then u is homotopic to v in the respective set, to show that the inclusion is a pi_0-equivalence.)

• Page 213, l. -7: Remove the equation "s(v(t)diag(z(t)*, z(t))) = 1_2n". (This equation does not make sense because we are not working in a C*-algebra with an adjoined unit, and the equation is not needed for anything.)

• Page 220, Eq. (13.1): Replace two occurances of the interval "[0,2pi]" with "[0,1]".

Clarifications and minor corrections:

• Page 2, l. -1: Insert ``(a quotient of)'' in front of ``A as a vector space...''.

• Page 3, l. 2: After "... to this inner product." add the following text "(It requires extra work to make phi injective, and this is often done by taking the infinite direct sum of all such Hilbert spaces, one Hilbert space for each positive linear functional on A.)".

• Page 9, l. -2: Remove one ``the''.

• Page 17: After line 7 add "because the unitary on the left-hand side has spectrum {-1,1} which is a strict subset of the circle T."

• Page 22, l. 10: Replace this line by "|alpha|=1, that u is unitary, and that q = upu*."

• Page 35, l. -4: After "Let (S,+) be an Abelian semigroup" add ", not necessarily with a neutral element. (We have chosen to work in this generality although the semigroups we shall consider actually do have a neutral element)."

• Page 36, l. 9: After "It is called the Grothendieck map." add "If S has a neutral element 0, then gamma_S is given by the simpler formula gamma_S(x)= <x, 0>."

• Page 67, l. -11: To see the second last equality, use Lemma 4.3.1 (ii).

• Page 68, l. 8: Most standard text books on algebraic topology contain the Five Lemma; see for example (14.7) in M. J. Greenberg and J. R. Harper Algebraic topology, Addison-Wesley, 1981.

• Page 137, l. 16: After "Lemma 2.1.3 (ii)" add "(or Corollary 2.1.4)".

• Page 140, l. 13: Replace Lemma 8.2.3 (i) by the slightly more precise: "there is a unitary u in U_n(A^~) for some n such that g = [u]₁ and phi^~(u) ~_h 1 in U_n(B^~).

• Page 177, l. -13: Before "Each element g in ...'' add ``The two unitaries u and v referred to in the theorem do exist as we shall proceed to show.''

• Page 177, l. -10: After "Lemma 2.1.3 (ii)" add "(or Corollary 2.1.4)".

• Page 198, l. 18: Use the Whitehead Lemma (Lemma 2.1.5) and its proof to see the last homotopy.