Universal arrows to forgetful functors from categories of topological algebra

Abstract

We survey the present trends in theory of universal arrows to forgetful functors from various categories of topological algebra and functional analysis to categories of topology and topological algebra. Among them are free topological groups, free locally convex spaces, free Banach-Lie algebras, and more. An accent is put on the relationship of those constructions with other areas of mathematics and their possible applications. A number of open problems is discussed; some of them belong to universal arrow theory, and other may become amenable to the methods of this theory.

Free Abelian topological groups and the Pontryagin-Van Kampen duality

“…The decision to publish the results now, more than a decade later, was motivated by an unceasing, yet moderate, flow of research on Pontryagin-van Kampen duality (see [3] or the recent elegant notes [27] and [15]). In general terms, dual objects, of which the character groups form a simple commutative speciman, are of paramount importance in quantum group theory and noncommutative analysis and geometry; on the other hand, various close relatives of free Abelian topological groups seem to gain in significance as well (see our recent survey [23]).…”

Notes on non-archimedean topological groups

“…Considering the category of uniform spaces and uniformly continuous maps one obtains the definition of a uniform free topological group F (X, U ) (see [39]). A description of the topology of this group was given by Pestov [40,41]. Free topological G-groups, the G-space version of the above notions, were introduced in [31].…”

Operator Spaces and Residually Finite-Dimensional C*-Algebras

“…Sinceη is continuous and polynomials of the form q are dense in C * X , the essential space ofη is V , that is, the representationη is finite-dimensional. This was announced (without a proof) in our survey [17]. [12]).…”