Normed representations of weight quivers

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 ${\mathcal{A}}_{\ddot{I}}^p‚$ in this paper whose object is a triple $(N,v,\hat{I}´)$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $\hat{I}´:N^{{\oplus }_p{2}^{\dim A}}\to N$ and each morphism $(N,v,\hat{I}´)\to (N^{\prime },v^{\prime },\hat{I}{´}^{\prime })$ is given by an $(A,B)$-homomorphism $\hat{I}¸:N\to N^{\prime }$ such that $\hat{I}¸(v)=v^{\prime }$ and $\hat{I}{´}^{\prime }\hat{I}{¸}^{\oplus 2^{\dim A}}=\hat{I}¸\hat{I}´$ hold. We show that ${\mathcal{A}}_{\ddot{I}}^p‚$ 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 ${\mathcal{A}}_{\ddot{I}}^p‚$ starting with this initial object.

50 pages, 5 figures