Nets of graded $C^{\ast }$-algebras over partially ordered sets

Abstract

The paper deals with $C^{\ast }$-algebras generated by a net of Hilbert spaces over a partially ordered set. The family of those algebras constitutes a net of $C^{\ast }$-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 $C^{\ast }$-algebras and their limits over maximal directed subsets. We also study properties of morphisms for nets of Hilbert spaces as well as nets of $C^{\ast }$-algebras.

20 pages