Let $X$ be a Banach space. We give characterizations of when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X)$ for every Banach space $Y$ in terms of nets of finite rank operators approximating weakly compact operators. Similar characterizations are given for the cases when $ℱ(X,Y)$ is a $u$-ideal in $𝒲(X,Y)$ for every Banach space $Y$, when $ℱ(Y,X)$ is a $u$-ideal in $𝒲(Y,X^{**})$ for every Banach space $Y$, and when $ℱ(Y,X)$ is a $u$-ideal in $𝒦(Y,X^{**})$ for every Banach space $Y$.

Let χ(m,n) be the unconditional basis constant of the monomial basis $z^{\alpha }$, α ∈ ℕ₀ⁿ with |α| = m, of the Banach space of all m-homogeneous polynomials in n complex variables, endowed with the supremum norm on the n-dimensional unit polydisc ⁿ. We prove that the quotient of $su{p}_m\sqrt[m]{su{p}_m\chi (m,n)}$ and √(n/log n) tends to 1 as n → ∞. This reflects a quite precise dependence of χ(m,n) on the degree m of the polynomials and their number n of variables. Moreover, we give an analogous formula for m-linear forms, a reformulation...

Let X be a Banach space. We study the circumstances under which there exists an uncountable set 𝓐 ⊂ X of unit vectors such that ||x-y|| > 1 for any distinct x,y ∈ 𝓐. We prove that such a set exists if X is quasi-reflexive and non-separable; if X is additionally super-reflexive then one can have ||x-y|| ≥ slant 1 + ε for some ε > 0 that depends only on X. If K is a non-metrisable compact, Hausdorff space, then the unit sphere of X = C(K) also contains such a subset; if moreover K is perfectly...

We prove uniform factorization results that describe the factorization of compact sets of compact and weakly compact operators via Hölder continuous homeomorphisms having Lipschitz continuous inverses. This yields, in particular, quantitative strengthenings of results of Graves and Ruess on the factorization through ${\ell }_p$-spaces and of Aron, Lindström, Ruess, and Ryan on the factorization through universal spaces of Figiel and Johnson. Our method is based on the isometric version of the Davis-Figiel-Johnson-Pełczyński...

In a Banach algebra an invertible element which has norm one and whose inverse has norm one is called unitary. The algebra is unitary if the closed convex hull of the unitary elements is the closed unit ball. The main examples are the C*-algebras and the ℓ₁ group algebra of a group. In this paper, different characterizations of unitary algebras are obtained in terms of numerical ranges, dentability and holomorphy. In the process some new characterizations of C*-algebras are given.

We show that if a separable Banach space X contains an isometric copy of every strictly convex separable Banach space, then X contains an isometric copy of l1 equipped with its natural norm. In particular, the class of strictly convex separable Banach spaces has no universal element. This provides a negative answer to a question asked by J. Lindenstrauss.

Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T:L\left(f\right)\equiv \overline{span}f\left(X\right)\to X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally right-stable...