Scaled-free objects II

TL;DR

This paper develops new categories of array-weighted sets to construct universal matrix-normed spaces and algebras, enabling the proof of free products for matrix-normed algebras via algebraic methods.

Contribution

It introduces two categories of array-weighted sets and constructs universal matrix-normed spaces and algebras with a universal property similar to free vector spaces.

Findings

Established universal properties for matrix-normed spaces and algebras.

Proved the existence of free products for matrix-normed algebras.

Provided algebraic constructions for these universal objects.

Abstract

This work creates two categories of "array-weighted sets" for the purposes of constructing universal matrix-normed spaces and algebras. These universal objects have the analogous universal property to the free vector space, lifting maps completely bounded on a generation set to a completely bounded linear map of the matrix-normed space. Moreover, the universal matrix-normed algebra is used to prove the existence of a free product for matrix-normed algebras using algebraic methods.