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.
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Funding
The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project no. 1.13556.2019/13.1.
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Gumerov, R.N., Lipacheva, E.V. Inductive Systems of \(\boldsymbol{C}^{\boldsymbol{*}}\)-Algebras over Posets: A Survey. Lobachevskii J Math 41, 644–654 (2020). https://doi.org/10.1134/S1995080220040137
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DOI: https://doi.org/10.1134/S1995080220040137