We study Markov measures and p-adic random walks with the use of states on the Cuntz algebras Op. Via the Gelfand–Naimark–Segal construction, these come from families of representations of Op. We prove that these representations reflect selfsimilarity especially well. In this paper, we consider a Cuntz–Krieger type algebra where the adjacency matrix depends on a parameter q ( q=1 is the case of Cuntz–Krieger algebra). This is an ongoing work generalizing a construction of certain measures associated to random walks on graphs.

A Banach space is an Asplund space if every continuous gauge has a point where the subdifferential mapping is Hausdorff weak upper semi-continuous with weakly compact image. This contributes towards the solution of a problem posed by Godefroy, Montesinos and Zizler.

We define and study λ-strict ideals in Banach spaces, which for λ=1 means strict ideals. Strict u-ideals in their biduals are known to have the unique ideal property; we prove that so also do λ-strict u-ideals in their biduals, at least for λ>1/2. An open question, posed by Godefroy et al. [‘Unconditional ideals in Banach spaces’, Studia Math.104 (1993), 13–59] is whether the Banach space X is a u-ideal in Ba(X), the Baire-one functions in X**, exactly when κu(X)=1; we prove that if κu(X)=1then X is a strict u-ideal in Ba (X) , and we establish the converse in the separable case.

Some of the results of § 5 of the cited paper are incorrect: in particular, the characterization of when an algebra is ultra-amenable, in terms of a diagonal like construction, is not proved; and Theorem 5.7 is stated wrongly. The rest of the paper is unaffected. We shall show in this corrigendum that Theorem 5.7 can be corrected and that the other results of § 5 are true if the algebra in question has a certain approximation property.

We exhibit a real Banach space M such that C(K,M) is almost transitive if K is the Cantor set, the growth of the integers in its Stone–Čech compactification or the maximal ideal space of L∞. For finite K, the space C(K,M) = M|K| is even transitive.

Given a separable Banach space E, we construct an extremely non-complex Banach space (i.e. a space satisfying that ‖ Id + T2 ‖ = 1 + ‖ T2 ‖ for every bounded linear operator T on it) whose dual contains E* as an L-summand. We also study surjective isometries on extremely non-complex Banach spaces and construct an example of a real Banach space whose group of surjective isometries reduces to ±Id, but the group of surjective isometries of its dual contains the group of isometries of a separable infinite-dimensional Hilbert space as a subgroup.

Let X and Y be separable Banach spaces and denote by 𝒮𝒮(X,Y ) the subset of ℒ(X,Y ) consisting of all strictly singular operators. We study various ordinal ranks on the set 𝒮𝒮(X,Y ). Our main results are summarized as follows. Firstly, we define a new rank r𝒮 on 𝒮𝒮(X,Y ). We show that r𝒮 is a co-analytic rank and that it dominates the rank ϱ introduced by Androulakis, Dodos, Sirotkin and Troitsky [Israel J. Math.169 (2009), 221–250]. Secondly, for every 1≤p<+∞, we construct a Banach space Yp with an unconditional basis such that 𝒮𝒮(ℓp,Yp) is a co-analytic non-Borel subset of ℒ(ℓp,Yp) yet every strictly singular operator T:ℓp→Yp satisfies ϱ(T)≤2. This answers a question of Argyros.

It is shown that a separable Hilbert space can be covered by non-overlapping closed convex sets Ci with outer radii uniformly bounded from above and inner radii uniformly bounded from below. This answers a question originating from the work of Klee.

We consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.

Answering a question of W. Arendt and M. Kunze in the negative, we construct a Banach space X and a norm closed weak* dense subspace Y of the dual X′ of X such that X, endowed with the Mackey topology μ(X,Y ) of the dual pair 〈X,Y 〉, is not complete.

We prove that Alexandrov spaces of non-negative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of an Alexandrov space of non-negative curvature into a 2-uniformly convex Banach space can be extended to a Lipschitz continuous map on the entire space.

In this note, we provide an example of a Banach space X for which that is not isomorphic to any Hilbert space, where denotes the infimum of all von Neumann–Jordan constants for equivalent norms of X.

We define asymptotically p-flat and innerly asymptotically p-flat sets in Banach spaces in terms of uniform weak* Kadec–Klee asymptotic smoothness, and use these concepts to characterize weakly compactly generated (Asplund) spaces that are c0(ω1)-generated or ℓp(ω1)-generated, where p∈(1,∞). In particular, we show that every subspace of c0(ω1) is c0(ω1)-generated and every subspace of ℓp(ω1) is ℓp(ω1)-generated for every p∈(1,∞). As a byproduct of the technology of projectional resolutions of the identity we get an alternative proof of Rosenthal’s theorem on fixing c0(ω1).

Determining meaningful lower bounds on the supremal strict p-negative type of classes of finite metric spaces is a difficult nonlinear problem. In this paper we use an elementary approach to obtain the following result: given a finite metric space (X,d) there is a constant ζ>0, dependent only on n=∣X∣ and the scaled diameter 𝔇=(diamX)/min{d(x,y)∣x⁄=y} of X (which we may assume is >1), such that (X,d) has p-negative type for all p∈[0,ζ] and strict p-negative type for all p∈[0,ζ). In fact, we obtain A consideration of basic examples shows that our value of ζ is optimal provided that 𝔇≤2. In other words, for each 𝔇∈(1,2] and natural number n≥3, there exists an n-point metric space of scaled diameter 𝔇 whose supremal strict p-negative type is exactly ζ. The results of this paper hold more generally for all finite semi-metric spaces since the triangle inequality is not used in any of the proofs. Moreover, ζ is always optimal in the case of finite semi-metric spaces.

We characterize surjective nonexpansive mappings between unit spheres of ℒ∞(Γ)-type spaces. We show that such mappings turn out to be isometries and can be extended to linear isometries on the whole space ℒ∞(Γ).

We investigate the connection between measures of noncompactness of a bounded subset of a given Banach space and the corresponding measures of noncompactness of an ultrapower of this subset. The Kuratowski, Hausdorff and separation measures of noncompactness are considered. We prove that in the first two cases the measures of a subset are equal to the respective measures of ultrapowers of this subset. In the case of separation measure of noncompactness, the equality is not necessarily fulfilled.

We investigate the influence of interface conditions at a singularity of an indefinite canonical system on its Weyl coefficient. An explicit formula which parametrizes all possible Weyl coefficients of indefinite canonical systems with fixed Hamiltonian function is derived. This result is illustrated with two examples: the Bessel equation, which has a singular end point, and a Sturm–Liouville equation whose potential has an inner singularity, which arises from a continuation problem for a positive definite function.

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of is Arens regular, and give some evidence that this is so if and only if is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapower of a dual Banach algebraffi We study how tensor products of ultrapowers behave, and apply this to study the question of when every ultrapower of is amenable. We provide an abstract characterization in terms of something like an approximate diagonal, and consider when every ultrapower of a C*-algebra, or a group L1-convolution algebra, is amenable.

In this paper, we give a new characterization of reflexive Banach spaces in terms of the sum of two closed bounded convex sets.

An example is found of a nonreflexive Banach space X such that the union of {0} and the set of non-norm-attaining functionals on X contains no two-dimensional subspace.