CORRECTIONS ON THEORY OF OPERATOR ALGEBRAS VOLUME I

Masamichi Takesaki

Page 5, line ↑ 6:

x^{−1} = x_{0}^{−1}+∑_{n=0}^{∞}[x_{0}(x_{0} − x)]^{n}x_{0}^{−1}
⇓
x^{−1} = x_{0}^{−1}+∑_{n=1}^{∞}[x_{0}(x_{0} − x)]^{n}x_{0}^{−1}

Page 7, line ↑ 8:

(x − λ1)^{−1} = \frac{1}{λ} (\frac{1}{λ} − x)^{−1}
⇓
(x − λ1)^{−1} = \frac{1}{λ} (1 − \frac{x}{λ})^{−1}

Page 15, line ↑ 16

|ω(x)| ≤ lim_{n→∞}∥ω∥^{n}∥x^{n}∥^{\frac{1}{n}}
⇓
|ω(x)| ≤ lim_{n→∞}∥ω∥^{\frac{1}{n}} ∥x^{n}∥^{\frac{1}{n}}

Page 17, line ↑ 11, 12

∥x^{2n}∥ = ∥(x^{∗}x)^{2^{n−1}}∥^{2} = ⋯ = ∥x^{∗}x∥^{2^{n−1}}

⇓

∥x^{2n}∥ = ∥(x^{∗}x)^{2^{n−1}}∥ = ⋯ = ∥x^{∗}x∥^{2^{n−1}}

2 MASAMICHI TAKESAKI

Page 27, between line ↑ 1 and ↑ 3:

∥x(1 − u_{ε})^{1/2}∥ = ∥εx(h + ε)^{−1/2}∥
= ε∥(h + ε)^{−1/2}x^{∗}x(h + ε)^{−1/2}∥^{1/2}
= ε∥(h + ε)^{−1}h∥ ≤ ε
⇓
∥x(1 − u_{ε})^{1/2}∥ = ∥ε^{1/2}x(h + ε)^{−1/2}∥
= ε^{1/2}∥(h + ε)^{−1/2}x^{∗}x(h + ε)^{−1/2}∥^{1/2}
= ε^{1/2}∥(h + ε)^{−1}h∥ ≤ ε^{1/2}.

Page 28, line ↓ 1 :

∥x(1 − u_{ε})^{1/2}∥ ≤ ε ⇒ ∥x(1 − u_{ε})^{1/2}∥ ≤ ε^{1/2}.

Page 28, line ↓ 6 :

∥x(1 − u_{ε})^{1/2}∥ ≤ ε ⇒ ∥x(1 − u_{ε})^{1/2}∥ ≤ ε^{1/2}

Page 33, line↓ 2

𝔪_{Γ_{𝔪}} ⊃ 𝔪_{Γ} and Γ_{𝔪_{Γ}} ⊃ Γ
⇓
𝔪_{Γ_{𝔪}} ⊃ 𝔪 and Γ_{𝔪_{Γ}} ⊃ Γ

Page 39, line ↓ 8 :

lim ω((x^{∗}u_{i})^{∗})^{}¯ = ω(x^{∗})^{}¯
= \overline{ω(x)}
⇓
lim ω((x^{∗}u_{i})^{∗})^{}¯ = \overline{ω(x^{∗∗})}
= \overline{ω(x)}

Page 38, line ↑ 5 :

Ã ⇒ A_{I}

Page 38, line ↑ 3 :

ω((xa^{∗})^{∗}(xa)) ≥ 0 ⇒ ω((xa^{∗})^{∗}(xa^{∗})) ≥ 0

CORRECTIONS ON THEORY OF OPERATOR ALGEBRAS VOLUME I 3

Page 39, line ↑ 9 :

⋯ ≤ |ω (\frac{1}{2}(x + x^{∗})) + ⋯
⇓
⋯ ≤ |ω (\frac{1}{2}(x + x^{∗}))| + ⋯

Page52, line ↓ 13 :

{π_{2}, ℌ} ⇒ {π_{2}, ℌ_{2}}

Page 63, line ↑ 13 and ↑ 4 :

x ∈ 𝒞(ℌ) ⇒ x ∈ ℒ𝒞(ℌ)

Page 69, line ↑ 18 :

ω(x) = (xξ̃ | η̃), x ∈ ℒ(ℌ).
⇓
ω(x) = (x̃ξ̃ | η̃), x ∈ ℒ(ℌ).

Page 73, line ↑ 11 :

U_{i}^{∗}x̃U_{i} = U_{j}^{∗}xU_{j}. ⇒ U_{i}^{∗}x̃U_{i} = U_{j}^{∗}x̃U_{j}.

Page 75, line ↑ 4 :

(ℳ′_{e})′ = ℳ′_{e} ⇒ (ℳ_{e})′ = (ℳ′)_{e}

Page 81, line ↑ 9 :

⋯ − (k − i)(k + i)ξ∥ ⇒ ⋯ − (k − i)(k + i)^{−1}ξ∥

Department of Mathematics, University of California, Los Angeles, California 90095-1555