Corrections, clarifications, and up-dates to “An Introduction to K-theory for C^{∗}-algebras”

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 φ 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 the preimage, f^{−1}(K), of any compact subset K of X is compact.) In the second statement one must require that the image of φ is not a proper ideal of C_{0}(X). Page 11, l. -9: Replace “for all x, y in A, and...” with “for all x, y in Ã, and...”. Page 14, Exercise 1.14: Replace “if and only if” with “if”. Page 19, l. 12: Replace “retract” by “deformation retract”. Page 20, l. 4: Replace “each bounded subset Ω of A” by “each bounded subset Ω of A^{+}”. Page 41, l. -2: Replace “a collection of maps φ ↦ F (φ)” by “a map φ ↦ F (φ)” Page 46, 3.3.1: One should assume that the C^{∗}-algebra A is unital; at least in the last section, where K_{0}(τ) 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 v_{1}, v_{2}, …, v_{r} \underline{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 τ extends (possibly in a non-canonical way) to a continuous function τ: M_{2}(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): This statement should read:

Ker(μ_{n}) = {a ∈ A_{n} : lim_{m→∞} ∥φ_{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 “K_{0}(φ_{n,1}): K_{0}(A_{n}) → K_{0}(M_{n}(A))” with “K_{0}(φ_{n,1}): K_{0}(A) → K_{0}(M_{n}(A))”. Page 103, l. 4: Replace “...= K_{0}(g′) = g,” with “...= K_{0}(κ_{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} = [\underline{a}]_{1} in K_{1}(A).” Page 155, l. 11: Replace “and p in U_{2(n_{1}+n_{2})}(Ĩ) by” with “and p in P_{2(n_{1}+n_{2})}(Ĩ) by”. Page 167, l. -11: Replace ”an isomorphism” with ”injective” in ”Moreover, φ is \underline{an isomorphism} if and only if v...” Page 169, Eq. (9.13): Move the minus sign appearing in the matrix so that the equation reads:

v = (\begin{matrix}a & −(1 − aa^{∗})^{1/2} \\ (1 − a^{∗}a)^{1/2} & a^{∗} \end{matrix})

Page 188, l. 14: Correct “The inclusion GL_{n}(S̃A) ⊆ U_{n}(S̃A) is a π_{0}-equivalence...” to “The inclusion U_{n}(S̃A) ⊆ GL_{n}(S̃A) is a π_{0}-equivalence...” Page 188, l. 15: Replace “U_{n}(S̃A) ⊆ Inv_{0}(n) is also...” with “GL_{n}(S̃A) ⊆ Inv_{0}(n) is also...”. Page 195, l. -9: Replace two occurances of “1_{A}” with “1_{B}”. Page 196, l. 9-10: Correct “the inclusion P_{n}(A) ⊆ GI_{n}(A) is a π_{0}-equivalence:” to “the inclusion GI_{n}(A) ⊆ P_{n}(A) is a π_{0}-equivalence:”. Page 203, l. 6: Replace “retract” by “deformation retract”.

Page 203, l. 14–15: It is not true that

{u ∈ C([0, 1], V(A)) : u(0) = u(1) = 1} = {u ∈ U_{∞}(S̃A) : s(u) = 1},

but {u ∈ U_{∞}(S̃A) : s(u) = 1} is a dense subset of {u ∈ C([0, 1], V(A)) : u(0) = u(1) = 1}, and this inclusion is a π_{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 ∼_{h} v in the respective set, to show that the inclusion is a π_{0}-equivalence.)

Page 213, l. -7: Remove the equation

s(v(t) (\begin{matrix}z(t)^{∗} & 0 \\ 0 & z(t) \end{matrix}) ) = 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, 2π]” by “[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 φ 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} ⫋ 𝕋.”

Page 22, l. 10: Replace this line by “|α| = 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 γ_{S} is given by the simpler formula γ_{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}(Ã) for some n such that g = [u]_{1} and φ̃(u) ∼_{h} 1 in U_{n}(B̃),”

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.