Algebraic topology of $C^*$-algebras

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 $C^*$-algebra can be regarded as a generalization of locally compact, Hausdorff topological space $\mathcal X$. From the commutative commutative Gelfand-Na\u{\i}mark theorem it follows that the spectrum of any commutative $C^*$-algebra is a locally compact, Hausdorff space which have the exact information of the $C^*$-algebra. Here we consider a Gelfand spaces of $C^*$-algebras which can be regarded as a generalization of the spectrum. In case of commutative $C^*$-algebras the Gelfand space coincides with the spectrum. Generally Gelfand spaces are not Hausdorff and provide more detailed information of noncommutative $C^*$-algebras. Sometimes the Gelfand space contains the full information of noncommutative $C^*$-algebra. Usage of Gelfand spaces of $C^*$-algebra enables us to define some $C^*$-algebraic analogs of several notions of the classical algebraic topology.