Comments for my paper “An ordering for the Banach spaces”. Pacific J. Math. 108 (1983) 83–98.

• implicitly used in Prop. 15:

If ℨ ≺ 𝔜, then 𝔉(𝔛, 𝔜) ⊆ 𝔉(𝔛, ℨ). See [∗], below.

• actually proved in Prop. 3:

𝔉(𝔛, c_{0}) = {α ∈ 𝔛^{∗∗} : α is sequentially continuous on (𝔛^{∗}, w^{∗})}

• Wilansky, Mazur spaces, Internat. J. Math. & Math. Sci. 4 (1981) 39–53: A “μB-space” is a space satisfying the condition of Mazur. If 0 → 𝔜 → 𝔛 → ℨ → 0 is exact, 𝔜 reflexive, and ℨ is μB, then 𝔛 is μB.

• Kappeler, Banach spaces which are opposite to Grothendieck spaces: A “d-complete” space is a space satisfying the condition of Mazur. If 0 → 𝔜 → 𝔛 → ℨ → 0 is exact, 𝔜 and ℨ are d-complete, and B_{𝔛^{∗}} is w^{∗}-

sequentially compact, then 𝔛 is d-complete.

If 𝔛 ≺ c_{0} and B_{𝔛^{∗}} is sequentially compact, then L_{1}(μ, 𝔛) ≺ c_{0}.

• Tauberian operators:

Neidiger & Rosenthal, Longhorn Notes 82–83, pp. 211–221.

• J(η) ≺ J(ω) for η < ω_{1}.

• G. Emmanuele, A characterization of Banach spaces with the property (V^{∗}) of Pelczynski:

The complemented l^{1}-property of this paper is equivalent to “Pelczynski’s condition (V^{∗})”.

l^{1}(Γ), Γ measurable, is a counterexample to question (a).

• Talagrand’s counterexample:

Talagrand, A new type of affine Borel function, Math. Scand. 54 (1984) 183–188.

• Related:

Gonzales & Oneiva, Ideal functions and operator ideals.

[∗] ℨ ≺ 𝔜 ⟺ 𝔉(𝔛, 𝔜) ⊆ 𝔉(𝔛, ℨ) for all 𝔛.

• Riddle, An application of Edgar’s ordering of Banach spaces. In: Analysis at Urbana, Cambridge Univ. Press, 1989.

• Characterization:

Let 𝔛 and 𝔜 be Banach spaces. Then 𝔛 ≺ 𝔜 if and only if: any bounded net (x_{ξ}) in 𝔛 such that (T x_{ξ}) converges weakly in 𝔜 for all operators T : 𝔛 → 𝔜 must converge weakly in 𝔛.

Typeset by AMS-TEX

• Answer to a question in a letter from Riddle.

Write 𝔪_{2} for the least two-valued-measurable cardinal and 𝔠 for the cardinal of the continuum. If card Γ ≥ 𝔪_{2}, then l_{1}(Γ) ⊀ l_{∞}.

Proof. Let T : l_{1}(Γ) → l_{∞} be a bounded linear map. Since card(l_{∞}) = 𝔠, there is a set Γ_{1} of cardinality ≤ 𝔠 (we can take Γ_{1} = l_{∞}, for example) and a map θ : Γ → Γ_{1} so that T factors l_{1}(Γ) → l_{1}(Γ_{1}) → l_{∞}, say T = T̃ ∘ V, where V (e_{γ}) = e_{θ(γ)} on the standard unit vectors.

Now if α ∈ ca(Γ, 𝒫 (Γ)) is a 2-valued measure vanishing on singletons, then V^{∗∗}(α) ∈ ca(Γ_{1}, 𝒫 (Γ_{1})) is a 2-valued measure, hence V^{∗∗}(α) ∈ l_{1}(Γ_{1}). Thus T^{∗∗}(α) ∈ l_{∞}, so α ∈ 𝔉(l_{1}(Γ), l_{∞}). This shows that l_{1}(Γ) ⊀ l_{∞}.