Related papers: Haar Null Closed and Convex Sets in Separable Bana…

We introduce a novel notion of {\it local spectral gap} for general, possibly infinite, measure preserving actions. We establish local spectral gap for the left translation action $\Gamma\curvearrowright G$, whenever $\Gamma$ is a dense…

Let $E$ be a Banach space and $\X$ be the closed unit ball of the dual space $E^*$. For a compact set $K$ in $E$, we prove that $K$ is polynomially convex in $E$ if and only if there exist a unital commutative Banach algebra $A$ and a…

We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties.…

It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism…

In 1985 S.~Saeki and K.~Stromberg published the following question: {\it Does every infinite compact group have a subgroup which is not Haar measurable?} An affirmative answer is given for all compact groups with the exception of some…

If ${\mathcal C}\simeq 2^{\mathbb N}$ denotes the Cantor set realized as the infinite product of two-point groups, then a folklore result says the Cantor map from ${\mathcal C}$ into $[0,1]$ sends Haar measure to Lebesgue measure on the…

A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $\sigma$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for…

We prove that a closed convex subset $C$ of a complete linear metric space $X$ is polyhedral in its closed linear hull if and only if no infinite subset $A\subset X\backslash C$ can be hidden behind $C$ in the sense $[x,y]\cap C\not =…

The notion of Aronszajn-null sets generalizes the notion of Lebesgue measure zero on the real line and the Euclidean space to infinite dimensional Banach spaces. We present a game-theoretic approach to Aronszajn-null sets, establish its…

We show that, as conjectured by Adrien Douady back in 1972, every complete metric space is homeomorphic (moreover, isometric) to the locus of zeros of an analytic map between two Banach spaces. As a corollary, a paracompact topological…

We propose a new class of hypertopologies, called here weak$^{\ast }$ hypertopologies, on the dual space $\mathcal{X}^{\ast }$ of a real or complex topological vector space $\mathcal{X}$. The most well-studied and well-known hypertopology…

The concept of uniform convexity of a Banach space was generalized to linear operators between Banach spaces and studied by Beauzamy [1976]. Under this generalization, a Banach space X is uniformly convex if and only if its identity map I_X…

We give a class of bounded closed sets $C$ in a Banach space satisfying a generalized and stronger form of the Bishop-Phelps property studied by Bourgain in \cite{Bj} for dentable sets. A version of the {\it ``Bishop-Phelps-Bollob\'as"}…

The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual…

It is shown that if $S$ is a commuting family of weak$^{\ast }$ continuous nonexpansive mappings acting on a weak$^{\ast }$ compact convex subset $C$ of the dual Banach space $E$, then the set of common fixed points of $S$ is a nonempty…

The purpose of this note is to show that, if $\mcB$ is a uniformly convex Banach, then the dual space $\mcB'$ has a "Hilbert space representation" (defined in the paper), that makes $\mcB$ much closer to a Hilbert space then previously…

In the general theory of locally compact quantum groups, the notion of Haar measure (Haar weight) plays the most significant role. The aim of this paper is to carry out a careful analysis regarding Haar weight, in relation to general…

We generalize the notion of summable Szlenk index from a Banach space to an arbitrary weak$^*$-compact set. We prove that a weak$^*$-compact set has summable Szlenk index if and only if its weak$^*$-closed, absolutely convex hull does. As a…

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach…

We prove the following results: (i) Every absolutely weakly compact set in a Banach lattice is absolutely weakly sequentially compact. (ii) The converse of (i) holds if $E$ is separable or $B_{E^{**}}$ is absolutely weak$^*$ compact. (iii)…