A Bishop–Phelps–Bollobás type theorem for uniform algebras

This paper is devoted to showing that Asplund operators with range in a uniform Banach algebra have the Bishop–Phelps–Bollobás property, i.e., they are approximated by norm attaining Asplund operators at the same time that a point where the approximated operator almost attains its norm is approximated by a point at which the approximating operator attains it. To prove this result we use the weak^∗${}^{\ast }$-to-norm fragmentability of weak^∗${}^{\ast }$-compact subsets of the dual of Asplund spaces and we need to observe a Urysohn type result producing peak complex-valued functions in uniform algebras that are small outside a given open set and whose image is inside a Stolz region.     

K-theory and exact sequences of partial translation algebras

In an earlier paper, the authors introduced partial translation algebras as a generalisation of group C^?${\mathrm{C}}^{?}$-algebras. Here we establish an extension of partial translation algebras, which may be viewed as an excision theorem in this context. We apply this general framework to compute the K-theory of partial translation algebras and group C^?${\mathrm{C}}^{?}$-algebras in the context of almost invariant subspaces of discrete groups. This generalises the work of Cuntz, Lance, Pimsner and Voiculescu. In particular we provide a new perspective on Pimsner's calculation of the K-theory for a graph product of groups.     

A note on spectral triples and quasidiagonality

Spectral triples (of compact type) are constructed on arbitrary separable quasidiagonal C^*$C^{*}$-algebras. On the other hand an example of a spectral triple on a non-quasidiagonal algebra is presented.     

Stable orbit equivalence of Bernoulli actions of free groups and isomorphism of some of their factor actions

We give an elementary proof for Lewis Bowen’s theorem saying that two Bernoulli actions of two free groups, each having arbitrary base probability spaces, are stably orbit equivalent. Our methods also show that for all compact groups K$K$ and every free product Γ$\Gamma$ of infinite amenable groups, the factor Γ?K^Γ/K$\Gamma ?K^{\Gamma }/K$ of the Bernoulli action Γ?K^Γ$\Gamma ?K^{\Gamma }$ by the diagonal K$K$-action is isomorphic with a Bernoulli action of Γ$\Gamma$.     

A new condition for maximal monotonicity via representative functions

In this paper we give a weaker sufficient condition for the maximal monotonicity of the operator S+A^∗TA$S+A^{\ast }TA$, where S:X?X^∗$S:X?X^{\ast }$, T:Y?Y^∗$T:Y?Y^{\ast }$ are two maximal monotone operators, A:X→Y$A:X\to Y$ is a linear continuous mapping and X,Y$X,Y$ are reflexive Banach spaces. We prove that our condition is weaker than the generalized interior-point conditions given so far in the literature. This condition is formulated using the representative functions of the operators involved. In particular, we rediscover some sufficient conditions given in the past using the so-called Fitzpatrick function for the maximal monotonicity of the sum of two maximal monotone operators and for the precomposition of a maximal monotone operator with a linear operator, respectively.     

The Schwarz genus of the Stiefel manifold

In this paper we compute: the Schwarz genus of the Stiefel manifold _Vk(Rⁿ)$V_k({\mathbb{R}}^n)$ with respect to the action of the Weyl group _Wk:=(Z/2)^k?_Sk$W_k:={(\mathbb{Z}/2)}^k?{\mathfrak{S}}_k$, and the Lusternik–Schnirelmann category of the quotient space _Vk(Rⁿ)/_Wk$V_k({\mathbb{R}}^n)/W_k$. Furthermore, these results are used in estimating the number of critically outscribed parallelotopes around a strictly convex body, and Birkhoff–James orthogonal bases of a normed finite dimensional vector space.     

Star-generating vectors of Rudin's quotient modules

The purpose of this paper is to study a class of quotient modules of the Hardy module H²(Dⁿ)$H^2({\mathbb{D}}^n)$. Along with the two variables quotient modules introduced by W. Rudin, we introduce and study a large class of quotient modules, namely Rudin's quotient modules of H²(Dⁿ)$H^2({\mathbb{D}}^n)$. By exploiting the structure of minimal representations we obtain an explicit co-rank formula for Rudin's quotient modules.     

Compact imbeddings between variable exponent spaces with unbounded underlying domain

In this paper, we have established some compact imbedding theorems for some subspaces of W^1,p(x)(U)$W^{1,p\left(x\right)}\left(U\right)$ when the underlying domain U$U$ is unbounded. The domain we consider is mainly of type R^N(N≥2)$R^N(N\ge 2)$ or R^L×Ω(L≥2)$R^L\times \Omega (L\ge 2)$, where Ω⊂R^M$\Omega \subset R^M$ is a bounded domain with smooth boundary.     

Amenability and uniqueness

The main result of this paper is a characterization of properly infinite injective von Neumann algebras and of nuclear C^∗$C^{\ast }$-algebras by using a uniqueness theorem, based on generalizations of Voiculescu’s famous Weyl–von Neumann theorem.     

On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

Renormalized solutions for nonlinear elliptic equations with variable exponents and L data

We prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and L¹$L^1$ data. The functional setting involves Lebesgue–Sobolev space with variable exponents W^1,p(⋅)(Ω)$W^{1,p(\cdot )}\left(\Omega \right)$.     

Quantitative Dunford–Pettis property

We investigate possible quantifications of the Dunford–Pettis property. We show, in particular, that the Dunford–Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford–Pettis property. We prove that L¹$L^1$ spaces and C(K)$C\left(K\right)$ spaces possess both of them. We also show that several natural measures of weak non-compactness are equal in L¹$L^1$ spaces.