C*-Algebras over Topological Spaces: The Bootstrap Class

We carefully define and study C*-algebras over topological spaces, possibly non-Hausdorff, and review some relevant results from point-set topology along the way. We explain the triangulated category structure on the bivariant Kasparov theory over a topological space. We introduce and describe an analogue of the bootstrap class for C*-algebras over a finite topological space.

💡 Research Summary

The paper develops a systematic framework for C‑algebras that are “over” a topological space X, allowing X to be non‑Hausdorff, and then introduces a bootstrap class adapted to this setting. The authors begin by defining an X‑C‑algebra as a separable C‑algebra A equipped with a continuous map φ : Prim(A) → X from its primitive ideal spectrum to X. This map induces a family of closed ideals I_U = ⋂{p∈φ^{-1}(U)} p for each open set U⊂X, thereby decomposing A into fibres A_x = A / I{X{x}}. Morphisms between X‑C‑algebras are required to respect the underlying map φ, leading to the notion of X‑equivariant *‑homomorphisms.

With this notion in place, the authors construct the bivariant Kasparov theory KK(X). Objects are X‑C‑algebras, and morphisms are Kasparov (A,B)-modules that are X‑equivariant. They prove that KK(X) carries a natural triangulated‑category structure: the suspension ΣA is defined as C₀((0,1),A), mapping cones give the usual exact triangles, and any X‑equivariant short exact sequence 0→A→B→C→0 yields a distinguished triangle A→B→C→ΣA. The triangulated axioms are verified in the presence of the X‑equivariance constraint, showing that KK(X) is a robust homological tool for studying X‑C‑algebras.

The central contribution is the definition of a bootstrap class 𝔅(X) for finite spaces X. Starting from the classical bootstrap class 𝔅 (the smallest triangulated subcategory of separable, nuclear C‑algebras satisfying the UCT), the authors define 𝔅({}) = 𝔅 for the one‑point space. For a finite X they proceed recursively: for each point x∈X the closed ideal I_{ {x} } must belong to 𝔅({}); then any X‑C‑algebra A for which the extension 0→I_U→A→A/I_U→0 (with U a saturated open subset) has both ends in the already constructed bootstrap class is declared to lie in 𝔅(X). This yields the smallest triangulated subcategory of KK(X) that is closed under mapping cones, suspensions, direct sums, extensions, and inductive limits, and that contains all fibre algebras belonging to the classical bootstrap class.

Key properties of 𝔅(X) are established:

1. Triangulated closure – closed under suspensions, cones, and direct sums.
2. Extension closure – if in an X‑equivariant exact triangle two vertices belong to 𝔅(X), so does the third.
3. Inductive‑limit closure – countable directed colimits of objects in 𝔅(X) remain in 𝔅(X).
4. UCT and K‑theory computability – for A∈𝔅(X) each fibre A_x lies in the classical bootstrap class, so K‑theory can be computed fibrewise and the universal coefficient theorem holds for KK(X) with coefficients in K‑theory.

The authors compare 𝔅(X) with the classical bootstrap class. When X is Hausdorff and A is a continuous field of C‑algebras over X, the condition A∈𝔅(X) is equivalent to every fibre A_x belonging to the classical bootstrap class. In non‑Hausdorff situations the same equivalence holds after suitably controlling the “singular” points of X, showing that 𝔅(X) genuinely generalises the classical notion.

To illustrate the theory, several concrete examples are worked out:

• Alexandroff finite spaces (e.g., a two‑point Sierpiński space) where the bootstrap class can be described explicitly in terms of extensions of simple algebras.
• Skeletal spaces arising from posets, demonstrating how the recursive construction mirrors the order structure.
• Non‑regular spaces obtained as spectra of non‑commutative algebras, showing that the bootstrap class still captures nuclear, UCT‑satisfying algebras despite pathological topology.

These examples confirm that objects in 𝔅(X) admit tractable K‑theory calculations and that KK(X) morphisms between them can be classified using the UCT.

Finally, the paper discusses potential applications. The bootstrap class 𝔅(X) provides a natural setting for extending Elliott’s classification program to C‑algebras equipped with a prescribed primitive‑ideal map, i.e., “X‑equivariant” classification. It also suggests a pathway to develop non‑commutative topological invariants for spaces that are not Hausdorff, by using X‑equivariant K‑theory as a bridge between operator‑algebraic data and underlying point‑set topology. Future work may aim at extending the construction to infinite or non‑finite spaces, exploring connections with sheaf‑theoretic approaches, and applying the framework to dynamical systems where the orbit space is naturally non‑Hausdorff.