Parametrization of topological algebraic objects

Question

There are several results of the following form: if an algebraic objects is endowed with a topology (or rather uniformity) which is somehow compatible with the algebraic structure, this uniformity is given through a collection of "compatible" real-valued pseudometrics or pseudonorms.

Among the examples I can name the following:

• Pseudometrics on a uniform space with no algebraic structure;

• Pseudonorms on commutative topological groups;

• Quasinorms on topological vector spaces (I am not sure if I use these terms correctly);

• Seminorms on locally convex spaces;

• Riesz pseudonorms on topological vector lattices;

• Riesz seminorms on locally convex topological vector lattices;

• Submeasures on Boolean algebras (I am not sure about this one);

• Lattice pseudometrics on lattices.

Please give more examples in the comments.

Perhaps there is a general result of this kind in the "topological universal algebra":

Is there a theorem along the lines that if an algebraic structure with operations satisfying certain conditions (I will be happy if only constants, unary and binary operations are covered) is endowed with a uniformity which satisfies certain conditions, this uniformity is induced by pseudometrics of a certain "algebraically nice" type?

Answer 1

Please regard this as a tentative answer since I am not sure that it corresponds to the spirit of your query. Many of the situations that you use to illustrate your query can be coded by using the concept of pro-categories (Grothendieck et al). Thus the category of locally convex spaces (more precisely, the complete lcs´s) is closely related to the pro-category associated with that of Banach spaces. (General remark: in the following we shall tacitly assume that all spaces are complete in order to simplify the exposition--the transition to accommodating non-complete ones is standard). Thus if we start with $Ban$, the category of Banach spaces, we get that of complete) lcs´s, $Ban_1$, the same but with linear contractions as morphisms, the category of Saks spaces. We can carry on the same programme with various seed categories to obtain a rich variety of new categories, some familiar, some not yet investigated. Particularly fruitful as seed categories are Waelbroeck spaces (Buchwalter), Banach lattices, Banach algebras, $C^\ast$-algebras (the latter two commutative or non-commutative, with or without units) as a (small) sample.

Similarly, the category of metric spaces leads to that of uniform spaces.

We remark that there is a dual situation involving ind-categories which unifies the transition from say, Banach spaces (algebras, lattices,...) to various categories of bornological spaces (in the sense of Waelbroeck, Buchwalter and Hogbe-Nlend), resp. from compact spaces to compactological ones (Waelbroeck, Buchwalter). The duality between the ind- and pro- constructions can be formalised and provides a unified approach to many useful generalisations of the central dualities of abstract functional analysis (dualities between Banach spaces, between Banach and Waelbroeck spaces, between compact spaces and Banach algebras--Gelfand-Naimark, between $C(K)$ spaces and spaces of Radon measures--Riesz representation theorems,...).