Inductive Systems of $$\boldsymbol{C}^{\boldsymbol{*}}$$-Algebras over Posets: A Survey

Abstract

We survey the research on the inductive systems of $$C^{*}$$ -algebras over arbitrary partially ordered sets. The motivation for our work comes from the theory of reduced semigroup $$C^{*}$$ -algebras and local quantum field theory. We study the inductive limits for the inductive systems of Toeplitz algebras over directed sets. The connecting $$\ast$$ -homomorphisms of such systems are defined by sets of natural numbers satisfying some coherent property. These inductive limits coincide up to isomorphisms with the reduced semigroup $$C^{*}$$ -algebras for the semigroups of non-negative rational numbers. By Zorn’s lemma, every partially ordered set $$K$$ is the union of the family of its maximal directed subsets $$K_{i}$$ indexed by elements of a set $$I$$ . For a given inductive system of $$C^{*}$$ -algebras over $$K$$ one can construct the inductive subsystems over $$K_{i}$$ and the inductive limits for these subsystems. We consider a topology on the set $$I$$ . It is shown that characteristics of this topology are closely related to properties of the limits for the inductive subsystems.