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Inductive systems of \(C^\ast\)-algebras over posets: a survey. (English) Zbl 1450.81053

Summary: 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.

MSC:

81T05 Axiomatic quantum field theory; operator algebras
46L05 General theory of \(C^*\)-algebras
16G20 Representations of quivers and partially ordered sets
46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras
06F05 Ordered semigroups and monoids
46M40 Inductive and projective limits in functional analysis
47L80 Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
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