We study Banach spaces with directionally asymptotically controlled ellipsoid-approximations of the unit ball in finite-dimensional sections. Here these ellipsoids are the unique minimum volume ellipsoids, which contain the unit ball of the corresponding finite-dimensional subspace. The directional control here means that we evaluate the ellipsoids by means of a given functional of the dual space. The term 'asymptotical' refers to the fact that we take 'lim sup' over finite-dimensional subspaces. ...

Disjointification inequalities are proven for arbitrary martingale difference sequences and conditionally independent random variables of the form ${f_k\left(s\right)x_k\left(t\right)}_{k=1}ⁿ$, where $f_k$’s are independent and xk’s are arbitrary random variables from a symmetric space X on [0,1]. The main results show that the form of these inequalities depends on which side of L₂ the space X lies on. The disjointification inequalities obtained allow us to compare norms of sums of martingale differences and non-negative random variables with...

We compute the completely bounded Banach-Mazur distance between different finite-dimensional homogeneous Hilbertian operator spaces.

If X is a Banach space and C a convex subset of X*, we investigate whether the distance $d̂({\overline{co}}^{w*}\left(K\right),C):=supinf\left|\right|k-c\left|\right|:c\in C:k\in {\overline{co}}^{w*}\left(K\right)$ from ${\overline{co}}^{w*}\left(K\right)$ to C is M-controlled by the distance d̂(K,C) (that is, if $d̂({\overline{co}}^{w*}\left(K\right),C)\le Md̂(K,C)$ for some 1 ≤ M < ∞), when K is any weak*-compact subset of X*. We prove, for example, that: (i) C has 3-control if C contains no copy of the basis of ℓ₁(c); (ii) C has 1-control when C ⊂ Y ⊂ X* and Y is a subspace with weak*-angelic closed dual unit ball B(Y*); (iii) if C is a convex subset of X and X is considered canonically embedded into...

Let E be a Banach space and let $ℬ₁\left(B_{E*}\right)$ and $₁\left(B_{E*}\right)$ denote the space of all Baire-one and affine Baire-one functions on the dual unit ball $B_{E*}$, respectively. We show that there exists a separable L₁-predual E such that there is no quantitative relation between $dist(f,ℬ₁\left(B_{E*}\right))$ and $dist(f,₁\left(B_{E*}\right))$, where f is an affine function on $B_{E*}$. If the Banach space E satisfies some additional assumption, we prove the existence of some such dependence.

The results of the first part concern the existence of higher order ℓ₁ spreading models in asymptotic ℓ₁ Banach spaces. We sketch the proof of the fact that the mixed Tsirelson space T[(ₙ,θₙ)ₙ], ${\theta }_{n+m}\ge \theta ₙ\theta ₘ$ and $li{m}_n\theta {ₙ}^{1/n}=1$, admits an $\ell {₁}^{\omega }$ spreading model in every block subspace. We also prove that if X is a Banach space with a basis, with the property that there exists a sequence (θₙ)ₙ ⊂ (0,1) with $li{m}_n\theta {ₙ}^{1/n}=1$, such that, for every n ∈ ℕ, $\left|\right|{\sum }_{k=1}^m{x}_k\left|\right|\ge \theta ₙ{\sum }_{k=1}^m\left|\right|x_k\left|\right|$ for every ₙ-admissible block sequence ${\left(x_k\right)}_{k=1}^m$ of vectors in X, then there exists c > 0 such...

An approximation property of divergent sequences in normed vector spaces is discussed.

The above question was raised by Teodor Przymusiński in May, 1983, in an unpublished manuscript of his. Later on, it was recognized by Takao Hoshina as a question that is of fundamental importance in the theory of rectangular normality. The present paper provides a complete affirmative solution. The technique developed for the purpose allows one to answer also another question of Przymusiński's.

Given a positive Banach-Saks operator T between two Banach lattices E and F, we give sufficient conditions on E and F in order to ensure that every positive operator dominated by T is Banach-Saks. A counterexample is also given when these conditions are dropped. Moreover, we deduce a characterization of the Banach-Saks property in Banach lattices in terms of disjointness.

We consider majorization problems in the non-commutative setting. More specifically, suppose E and F are ordered normed spaces (not necessarily lattices), and 0 ≤ T ≤ S in B(E,F). If S belongs to a certain ideal (for instance, the ideal of compact or Dunford-Pettis operators), does it follow that T belongs to that ideal as well? We concentrate on the case when E and F are C*-algebras, preduals of von Neumann algebras, or non-commutative function spaces. In particular, we show that, for C*-algebras...