It is shown that if α,ζ are ordinals such that 1 ≤ ζ < α < ζω, then there is an operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself such that if Y is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$, then the operator is not an isomorphism on Y. This contrasts with a result of J. Bourgain that implies that there are uncountably many ordinals α for which for any operator from $C\left({\omega }^{{\omega }^{\alpha }}\right)$ onto itself there is a subspace of $C\left({\omega }^{{\omega }^{\alpha }}\right)$ which is isomorphic to $C\left({\omega }^{{\omega }^{\alpha }}\right)$ on which the operator is an isomorphism.

Let S¹ be the stopping time space and ℬ₁(S¹) be the Baire-1 elements of the second dual of S¹. To each element x** in ℬ₁(S¹) we associate a positive Borel measure ${\mu }_{x**}$ on the Cantor set. We use the measures ${\mu }_{x**}:x**\in ℬ₁\left(S¹\right)$ to characterize the operators T: X → S¹, defined on a space X with an unconditional basis, which preserve a copy of S¹. In particular, if X = S¹, we show that T preserves a copy of S¹ if and only if ${\mu }_{T**(x**)}:x**\in ℬ₁\left(S¹\right)$ is non-separable as a subset of $ℳ\left(2^{\mathbb{N}}\right)$.

For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xₙ) in X with $K\subseteq \sum ₙ\alpha ₙxₙ:\left(\alpha ₙ\right)\in B_{{\ell }_{p^{\text{'}}}}$. We prove that an operator T: X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T* is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.

Two of James’ three quasi-reflexive spaces, as well as the James Tree, have the uniform $w^{*}$-Opial property.

Suppose E is fully symmetric Banach function space on (0,1) or (0,∞) or a fully symmetric Banach sequence space. We give necessary and sufficient conditions on f ∈ E so that its orbit Ω(f) is the closed convex hull of its extreme points. We also give an application to symmetrically normed ideals of compact operators on a Hilbert space.

Let T be a bounded linear operator on a (real or complex) Banach space X. If (aₙ) is a sequence of non-negative numbers tending to 0, then the set of x ∈ X such that ||Tⁿx|| ≥ aₙ||Tⁿ|| for infinitely many n’s has a complement which is both σ-porous and Haar-null. We also compute (for some classical Banach space) optimal exponents q > 0 such that for every non-nilpotent operator T, there exists x ∈ X such that $\left(\right|\left|Tⁿx\right||/\left|\right|Tⁿ\left|\right|)\notin {\ell }^q\left(\mathbb{N}\right)$, using techniques which involve the modulus of asymptotic uniform smoothness of X.

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We give some criteria for order boundedness of $E\left(\mu \right)$ in $ba\left(\Re \right)$, in the general case as well as for atomic $\mu$. Order boundedness implies weak compactness of $E\left(\mu \right)$. We show that the converse implication holds under some assumptions on $𝔐$, $\Re$ and $\mu$ or $\mu$ alone, but not in general.

The space of all order continuous linear functionals on an Orlicz space $L^{\varphi }$ defined by an arbitrary (not necessarily convex) Orlicz function $\varphi$ is described.

We study order convexity and concavity of quasi-Banach Lorentz spaces ${\Lambda }_{p,w}$, where 0 < p < ∞ and w is a locally integrable positive weight function. We show first that ${\Lambda }_{p,w}$ contains an order isomorphic copy of $l^p$. We then present complete criteria for lattice convexity and concavity as well as for upper and lower estimates for ${\Lambda }_{p,w}$. We conclude with a characterization of the type and cotype of ${\Lambda }_{p,w}$ in the case when ${\Lambda }_{p,w}$ is a normable space.

Let $K$ be a compact space and let $C\left(K\right)$ be the Banach lattice of real-valued continuous functions on $K$. We establish eleven conditions equivalent to the strong compactness of the order interval $[0,x]$ in $C\left(K\right)$, including the following ones: (i) $\{x>0\}$ consists of isolated points of $K$; (ii) $[0,x]$ is pointwise compact; (iii) $[0,x]$ is weakly compact; (iv) the strong topology and that of pointwise convergence coincide on $[0,x]$; (v) the strong and weak topologies coincide on $[0,x]$. Moreover, the weak topology and that of pointwise convergence...

In earlier papers we have introduced and studied a new notion of positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. Here we continue to develop this positivity and its associated ordering, proving many foundational facts. We also give many applications, for example to noncommutative topology, noncommutative peak sets, lifting problems, peak interpolation, approximate identities, and to order relations between an operator...

Let $𝔐$ and $\Re$ be algebras of subsets of a set $\Omega$ with $𝔐\subset \Re$, and denote by $E\left(\mu \right)$ the set of all quasi-measure extensions of a given quasi-measure $\mu$ on $𝔐$ to $\Re$. We show that $E\left(\mu \right)$ is order bounded if and only if it is contained in a principal ideal in $ba\left(\Re \right)$ if and only if it is weakly compact and $extrE\left(\mu \right)$ is contained in a principal ideal in $ba\left(\Re \right)$. We also establish some criteria for the coincidence of the ideals, in $ba\left(\Re \right)$, generated by $E\left(\mu \right)$ and $extrE\left(\mu \right)$.

We study the c₀-content of a seminormalized basic sequence (χₙ) in a Banach space, by the use of ordinal indices (taking values up to ω₁) that determine dichotomies at every ordinal stage, based on the Ramsey-type principle for every countable ordinal, obtained earlier by the author. We introduce two such indices, the c₀-index ${\xi }^{\left(\chi ₙ\right)}₀$ and the semibounded completeness index ${\xi }_b^{\left(\chi ₙ\right)}$, and we examine their relationship. The countable ordinal values that these indices can take are always of the form ${\omega }^{\zeta }$. These results...