Nets of graded $C^*$-algebras over partially ordered sets
Suren Arshakovich Grigoryan, Ekaterina Vladimirovna Lipacheva, Аirat Sitdikov · St Petersburg Mathematical Journal · 2019
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 algebra is graded by the first homotopy group of the partially ordered set. Inductive systems of $C^*$-algebras and their limits over maximal directed subsets are considered, as well as properties of morphisms for nets of Hilbert spaces and nets of $C^*$-algebras.