Nets of graded $C^*$-algebras over partially ordered sets

TL;DR

This paper explores the structure of $C^*$-algebras generated by nets of Hilbert spaces over partially ordered sets, revealing their grading by the first homotopy group and analyzing inductive limits and morphisms.

Contribution

It introduces a novel grading of $C^*$-algebras by the first homotopy group of the underlying set and studies their inductive limits and morphisms.

Findings

Every such algebra is graded by the first homotopy group.

Analysis of inductive systems and their limits over maximal directed subsets.

Properties of morphisms for nets of Hilbert spaces and $C^*$-algebras.

Abstract

The paper deals with $C^*$-algebras generated by a net of Hilbert spaces over a partially ordered set. The family of those algebras constitutes a net of $C^*$-algebras over the same set. It is shown that every such an algebra is graded by the first homotopy group of the partially ordered set. We consider inductive systems of $C^*$-algebras and their limits over maximal directed subsets. We also study properties of morphisms for nets of Hilbert spaces as well as nets of $C^*$-algebras.