A Unified Approach to the Arens Regularity and Related Problems for a Class of Banach Algebras Associated With Locally Compact Groups

Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.

Spatial structure of national administrative division’s systems

The spatial parameters of systems of administrative-territorial division (ATD) are considered: contact number (neighbors number), transportation permeability (average distance between neighbor intersection of administrative border by transportation routes), shapes of administrative-territorial units (some indices, including Morton index), graph theoretical indexes of Karel Kansky described the peculiarities of topological structures of ATD systems (connectivity and form). Main concepts of graph theory which are applied to describe the level of spatial (topological) complexity of ATD network are explained in detail: vertices, edges, circuit (closed round), vertices degree, topological center, circuital framework, topological tier. The technique of assignment of topological tiers in the circuital framework of ATD-system for the analysis of spatial structure of its network for 1st level of hierarchy of 9 nations is applied. Maximal number of topological tiers – 20 – for ATD system on 2nd level hierarchy is revealed on the example of the system of counties of U.S.A. The variation of numbers of topological tiers depending on the level of hierarchy of ATD is shown in the case of Spain: 2 – for 1st level hierarchy, 3 – for 2nd, 7 – for 3th, 24 – for 4th. 5 types of topological structure of national systems of ATD are distinguished: compact roundish, compact, elongated with particular distensions, elongated with parallel connections, linearly elongated with truncated circuital framework.

Amalgamated duplication of the Banach algebra 𝔄 along a 𝔄-bimodule 𝒜

Let [Formula: see text] and [Formula: see text] be Banach algebras such that [Formula: see text] is a Banach [Formula: see text]-bimodule with compatible actions. We define the product [Formula: see text], which is a strongly splitting Banach algebra extension of [Formula: see text] by [Formula: see text]. After characterization of the multiplier algebra, topological center, (maximal) ideals and spectrum of [Formula: see text], we restrict our investigation to the study of semisimplicity, regularity, Arens regularity of [Formula: see text] in relation to that of the algebras [Formula: see text], [Formula: see text] and the action of [Formula: see text] on [Formula: see text]. We also compute the first cohomology group [Formula: see text] for all [Formula: see text] as well as the first-order cyclic cohomology group [Formula: see text], where [Formula: see text] is the [Formula: see text]th dual space of [Formula: see text] when [Formula: see text] and [Formula: see text] itself when [Formula: see text]. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Lau products and module extensions of Banach algebras as well as triangular Banach algebra. Finally, special attention is devoted to the cyclic and [Formula: see text]-weak amenability of [Formula: see text]. In this context, several open questions arise.