Algebraic topology of $C^*$-algebras
Petr Ivankov

TL;DR
This paper explores the use of Gelfand spaces as a generalization of spectra for $C^*$-algebras, aiming to develop algebraic topology concepts applicable to noncommutative cases.
Contribution
It introduces the concept of Gelfand spaces for $C^*$-algebras as a tool to extend classical algebraic topology to noncommutative settings.
Findings
Gelfand spaces can contain full information of noncommutative $C^*$-algebras.
Gelfand spaces are not necessarily Hausdorff, providing richer structure.
The approach enables defining $C^*$-algebraic analogs of classical topology notions.
Abstract
Any -algebra can be regarded as a generalization of locally compact, Hausdorff topological space . From the commutative commutative Gelfand-Na\u{\i}mark theorem it follows that the spectrum of any commutative -algebra is a locally compact, Hausdorff space which have the exact information of the -algebra. Here we consider a Gelfand spaces of -algebras which can be regarded as a generalization of the spectrum. In case of commutative -algebras the Gelfand space coincides with the spectrum. Generally Gelfand spaces are not Hausdorff and provide more detailed information of noncommutative -algebras. Sometimes the Gelfand space contains the full information of noncommutative -algebra. Usage of Gelfand spaces of -algebra enables us to define some -algebraic analogs of several notions of the classical algebraic topology.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Homotopy and Cohomology in Algebraic Topology · Advanced Banach Space Theory
