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.

中文翻译：

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$$ \ boldsymbol {C} ^ {\ boldsymbol {*}} $$的归纳系统-Posets上的代数：一项调查

摘要

我们调查了关于（（C ^ {*} \））-代数在任意部分有序集上的归纳系统的研究。我们工作的动机来自于简化半群\（C ^ {*} \）-代数理论和局部量子场论。我们研究了有向集上Toeplitz代数的归纳系统的归纳极限。这样的系统的连接\（\ ast \）-同态由满足某些相干性质的自然数集定义。对于非负有理数的半群，这些归纳极限与减少的半群\（C ^ {*} \）-代数符合同构。通过Zorn引理，每个部分有序集\（K \）是其最大有向子集族的并集\（K_ {i} \）由一组\（I \）的元素索引。对于给定的\（C ^ {*} \）- \（K \）上的代数系统，可以构造一个\（K_ {i} \）上的归纳子系统以及这些子系统的归纳极限。我们考虑集合\（I \）上的拓扑。结果表明，这种拓扑的特性与感应子系统的极限特性密切相关。