
TL;DR
This paper introduces a new framework for constructing free normed objects in functional analysis, extending algebraic free object concepts to Banach algebras through scaled-free properties and adjoint functors.
Contribution
It develops a categorical approach to define scaled-free normed objects, enabling algebraic-like constructions in functional analytic settings.
Findings
Defined scaled-free normed objects via adjoint functors
Established universal properties for these scaled-free objects
Lays groundwork for presentation theory of Banach algebras
Abstract
In this work, I address a primary issue with adapting categorical and algebraic concepts to functional analytic settings, the lack of free objects. Using a "normed set" and associated categories, I describe constructions of normed objects, which build from a set to a vector space to an algebra, and thus parallel the natural progression found in algebraic settings. Each of these is characterized as a left adjoint functor to a natural forgetful functor. Further, the universal property in each case yields a "scaled-free" mapping property, which extends previous notions of "free" normed objects. In subsequent papers, this scaled-free property, coupled with the associated functorial results, will give rise to a presentation theory for Banach algebras and other such objects, which inherits many properties and constructions from its algebraic counterpart.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Advanced Banach Space Theory
