CORRECTIONS ON THEORY OF OPERATOR ALGEBRAS VOLUME II

Masamichi Takesaki

Page 8, line ↓ 16

“f(λ) = λg(λ)f(λ)” ⇒ “f(λ) = λg(λ)”

line ↓ 17

\begin{aligned}g(h)hf(h & ⇒ g(h)h \\ g(k)kf(k) & ⇒ g(k)k \end{aligned} i.e., Erase f(h) and f(k)

Page 12, line ↓ 14

𝔇(Δ^{− \frac{1}{2}} ) ∩ 𝔇(Δ^{− \frac{1}{2}} ) ⇒ 𝔇(Δ^{\frac{1}{2}} ) ∩ 𝔇(Δ^{− \frac{1}{2}} )

Page 49, line ↑ 7

ω_{η_{φ}}(x_{n}) ⇒ ω_{η_{φ}(x_{n})}

Page 59, line ↓ 13

(ℳ)_{∗}^{+} ⇒ (ℳ′)_{∗}^{+}

Page 60, The arguments between the line ↑ 11 and the line ↑ 9: “Since θ′(𝔪_{r}^{+}) ⊂ ℳ_{∗}^{+}, ω′_{a} is positive and

∥ω′_{a}∥ = ⋯ = λ < +∞.

Hence ω′_{a} is bounded, so that ⋯ by ω′_{a} again.”

⇓

“Since θ′(𝔪_{r}^{+}) ⊂ ℳ_{∗}^{+}, ω′_{a} is positive and

sup{ω′_{a}(y) : y ∈ 𝔪_{r}^{+} ∩ 𝒮′_{0}} = sup{⟨a, θ′(y)⟨: y ∈ 𝔪_{r}^{+} ∩ 𝒮′_{0}}

= sup{⟨a, ω⟩ : ω ∈ Φ_{ℓ,0}} = ψ(a) = λ < +∞.

Since the intersection 𝔪_{r} ∩𝒜 of 𝔪_{r} and any abelian von Neumann subalgebra 𝒜 is an ideal of 𝒜, the absolute value |h|, the positive part h_{+} and the negative part h_{−} of a self-adjoint element h ∈ 𝔪_{r} are all in 𝔪_{r}, so that

\begin{aligned}|ω′_{a}(h + ik)| & ≤ ω′_{a}(h_{+}) + ω′_{a}(h_{−}) + ω′_{a}(k_{+}) + ω′_{a}(k_{−}) \\ & ≤ 4λ∥h + ik∥ \end{aligned}

for every self adjoint h, k ∈ 𝔪_{r}. Hence ω′_{a} is bounded, so that it can be extended to the norm closure A_{r} of 𝔪_{r} as a positive linear functional, which will be denoted by ω′_{a} again. Then the norm ∥ω′_{a}∥ of ω′_{a} is precisely equal to λ.”

Page 253, line ↑ 12:

δ_{G}(r)^{\frac{1}{2}} ⇒ δ_{G}(s)^{\frac{1}{2}}

Page 363, line ↑ 4

Section 5, is devoted ⇒ Section 5 is devoted

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