Algebraic topology of $C^*$-algebras
Petr Ivankov
TL;DR
This work introduces a noncommutative generalization of fundamental topological invariants by constructing a Gelfand-type space $\mathfrak{Gelfand}(A)$ from lattices of closed left ideals and Pedersen’s ideal $K(A)$, recovering the classical spectrum when $A=C_0(\mathcal{X})$ and encoding non-Hausdorff base spaces in the noncommutative setting. It develops a robust notion of noncommutative coverings via $(A,\widetilde{A},G,\mathfrak{lift})$, with a discrete fundamental group $G$ and a universal covering when it exists, paralleling ordinary coverings in the commutative case where $\pi_1(\mathcal{X})$ arises. The paper further analyzes concrete instances: the case $A=\mathcal{K}(\mathcal{H})$ yields $\mathfrak{Gelfand}(A)\cong \mathbb{C}P(\mathcal{H})$, while Hausdorff blowing-up connects noncommutative spectra to base spaces; for continuous-trace algebras, the Gelfand space is tied to bundle data $E$ over the spectrum with a Dixmier–Douady invariant classifying $A$ up to isomorphism. Altogether, the framework provides a noncommutative topological viewpoint that unifies spectrum, coverings, and bundle-theoretic aspects of $C^*$-algebras, with concrete bridges to classical invariants.
Abstract
Any $C^*$-algebra can be regarded as a generalization of locally compact Hausdorff topological space. Here we consider a generalization of fundamental group and (co)homology theory. In result one has invariants of $C^*$-algebras such that: for any commutative $C^*$-algebra $A = C_0\left(\mathcal X \right)$ the invariants of $A$ coincide with the $\mathcal X$ ones, the theory is not trivial even for algebras having a one spectrum, e.g. containing one point only. Instead of $\mathcal X$ we consider the space $\mathcal Y$ which should not be Hausdorff. However a noncommutative $C^*$-algebra $A$ defines $\mathcal Y$ and vice-versa.