On a Topology and Limits for Inductive Systems of C ^∗ -Algebras over Partially Ordered Sets

Abstract

© 2019, Springer Science+Business Media, LLC, part of Springer Nature. Motivated by algebraic quantum field theory and our previous work we study properties of inductive systems of C ∗ -algebras over arbitrary partially ordered sets. A partially ordered set can be represented as the union of the family of its maximal upward directed subsets indexed by elements of a certain set. We consider a topology on the set of indices generated by a base of neighbourhoods. Examples of those topologies with different properties are given. An inductive system of C ∗ -algebras and its inductive limit arise naturally over each maximal upward directed subset. Using those inductive limits, we construct different types of C ∗ -algebras. In particular, for neighbourhoods of the topology on the set of indices we deal with the C ∗ -algebras which are the direct products of those inductive limits. The present paper is concerned with the above-mentioned topology and the algebras arising from an inductive system of C ∗ -algebras over a partially ordered set. We show that there exists a connection between properties of that topology and those C ∗ -algebras.