Related papers: Metric duality for Abelian groups

In this notebook, I present duality theory (or theories) of abelian groups with some categorical and categorical topological flavour. I consider writing this notebook as a longer-term project, and its current content and presentation is…

If an open subgroup of the group of the invertible measures on a LCA group is isometric to another, then the correspoinding underlying LCA groups are topologically isomorphic to each other.

In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that the constructive Gelfand duality can be extended into a duality between compact regular locales and unital abelian localic C*-algebras. In order to do so we…

The notion of acyclic matching property was provided by Losonczy and it was proved that torsion-free groups admit this property. In this paper, we introduce a duality of acyclic matching as a tool for classification of some Abelian groups,…

In this paper, we study the metric dimension of Cayley graphs. Specially, we present a complete characterization of Cayley graphs on Abelian groups whose metric dimension is two.

The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of Sobolev functions. This will be achieved without…

Abelian categories provide a self-dual axiomatic context for establishing homomorphism theorems (such as the isomorphism theorems and homological diagram lemmas) for abelian groups, and more generally, modules. In this paper we describe a…

In 2008, the author proposed a version of duality theory for (not necessarily, Abelian) complex Lie groups, based on the idea of using the Arens-Michael envelope of topological algebra and having an advantage over existing theories in that…

We give a positive answer to the question of Shkarin (\emph{On universal abelian topological groups}, Mat. Sb. 190 (1999), no. 7, 127-144) whether there exists a metrically universal abelian separable group equipped with invariant metric.…

The notions of quasiconvexity, Wright convexity and convexity for functions defined on a metric Abelian group are introduced. Various characterizations of such functions, the structural properties of the functions classes so obtained are…

In the present paper, we introduce a new concept of convexity which is generated by a family of endomorphisms of an Abelian group. In Abelian groups equipped with a translation invariant metric, we define the boundedness, the norm, the…

We incorporate metric data into the framework of Tannaka-Krein duality. Thus, for any group with left invariant metric, we produce a dual metric on its category of unitary representations. We characterize the conditions under which a…

We study groups that can be defined as Polish, pro-countable groups, as non-archimedean groups with an invariant metric or as quasi-countable groups, i.e., closed subdirect products of countable, discrete groups, endowed with the product…

In this article we extend the notion of metric measure spaces to so-called metric two-level measure spaces (m2m spaces): An m2m space $(X, r, \nu)$ is a Polish metric space $(X, r)$ equipped with a two-level measure $\nu \in…

We construct a multiset space $\mathbb{N}[X]$ over a metric space $X$ that simultaneously enjoys desirable topological properties and admits a natural matching metric $d_{\mathbb{N}[X]}$, making it a metrizable abelian topological monoid…

We suggest a new generalization of Pontryagin duality from the category of Abelian locally compact groups to a category which includes all Moore groups, i.e. groups whose irreducible representations are finite-dimensional. Objects in this…

The Galilean Conformal Algebra (GCA) arises in taking the non-relativistic limit of the symmetries of a relativistic Conformal Field Theory in any dimensions. It is known to be infinite-dimensional in all spacetime dimensions. In…

We characterise Lie groups with bi-invariant bargmannian, galilean or carrollian structures. Localising at the identity, we show that Lie algebras with ad-invariant bargmannian, carrollian or galilean structures are actually determined by…

We examine sufficient conditions for the dual of a topological group to be metrizable and locally compact.

In this work, the dual flatness, which is connected with Statistics and Information geometry, of general $(\alpha,\beta)$-metrics (a new class of Finsler metrics) is studied. A nice characterization for such metrics to be dually flat under…