Inductive Systems of C *-Algebras over Posets: A Survey

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 *-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 subsetsKi indexed by elements of a set I. For a given inductive systemof C *-algebras over K one can construct the inductive subsystems overKi 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.