Normed representations of weight quivers

TL;DR

This paper introduces norms for tensor rings derived from weight quivers and constructs a category where various classical integration and approximation methods are represented as morphisms, revealing a deep algebraic structure.

Contribution

It defines a new category of bimodules with norms and special elements, integrating classical analysis tools as morphisms within this algebraic framework.

Findings

Existence of an initial object in the category.

Classical integration methods are morphisms in the category.

Approximation theorems are represented as morphisms.

Abstract

Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\mathscr{A}^p_{\varsigma}$ in this paper whose object is a triple $(N,v,\delta)$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $\delta: N^{\oplus_p 2^{\dim A}} \to N$ and each morphism $(N,v,\delta) \to (N',v',\delta')$ is given by an $(A,B)$-homomorphism $\theta: N\to N'$ such that $\theta(v)=v'$ and $\delta' \theta^{\oplus 2^{\dim A}} = \theta\delta$ hold. We show that $\mathscr{A}^p_{\varsigma}$ has an initial object such that Daniell integration, Bochner integration, Lebesgue integration, Stone--Weierstrass Approximation Theorem, power series expansion, and Fourier series expansion are morphisms in $\mathscr{A}^p_{\varsigma}$ starting with this initial object.