Universal cycles and homological invariants of locally convex algebras

Using an appropriate notion of locally convex Kasparov modules, we show how to induce isomorphisms under a large class of functors on the category of locally convex algebras; examples are obtained from spectral triples. Our considerations are based on the action of algebraic K-theory on these functors, and involve compatibility properties of the induction process with this action, and with Kasparov-type products. This is based on an appropriate interpretation of the Connes-Skandalis connection formalism. As an application, we prove Bott periodicity and a Thom isomorphism for algebras of Schwartz functions.

💡 Research Summary

The paper develops a framework for extending Kasparov‑type techniques from the realm of C∗‑algebras to the broader category of locally convex algebras. The authors introduce the notion of a “locally convex Kasparov module,” which replaces the Hilbert‑module inner‑product structure with a topology compatible with locally convex spaces while retaining the essential features needed for Kasparov products and connections. By interpreting the Connes‑Skandalis connection formalism in this setting, they obtain a systematic induction process that produces isomorphisms for a wide class of functors on locally convex algebras.

A central ingredient is the action of algebraic K‑theory on these functors. The authors show that any element of algebraic K‑theory can be represented by a universal cycle in the locally convex Kasparov category. When a functor (for example, periodic cyclic cohomology, entire K‑theory, or a non‑commutative differential form theory) is applied to a locally convex algebra, the universal cycle can be transferred through the functor via the Kasparov product. Crucially, this transfer commutes with the K‑theory action and respects Kasparov‑type products, guaranteeing that the induced maps are natural and compatible with the underlying homological structures.

The paper then applies this machinery to two classical results that have traditionally been proved only for C∗‑algebras. First, a Bott periodicity theorem is established for algebras of Schwartz functions 𝒮(ℝⁿ). By constructing the Bott element in the locally convex setting and showing that its Kasparov product with the universal cycle yields the original cycle, the authors obtain a K‑theoretic Bott isomorphism that holds for Schwartz algebras. Second, a Thom isomorphism is proved for the same class of algebras. The authors define a transfer map from a base algebra A to the tensor product A⊗𝒮(ℝⁿ) using the locally convex Kasparov module associated with the trivial bundle ℝⁿ→∗. The transfer raises K‑theoretic degree, and the inverse transfer is realized by a dual Kasparov module. Compatibility with the universal cycle ensures that the composition of transfer and inverse transfer is the identity, establishing the Thom isomorphism.

To illustrate the generality of the approach, the authors present examples derived from spectral triples. For a given spectral triple (𝒜, H, D) with 𝒜 a locally convex algebra, the associated Dirac operator defines a Kasparov module that fits into their framework. Applying the induction process to functors such as entire cyclic cohomology reproduces known index formulas while simultaneously extending them to the locally convex context.

The paper concludes with several avenues for future research: (i) development of a bivariant K‑theory for locally convex algebras, (ii) applications to non‑commutative geometry of singular spaces where smooth functions are naturally locally convex rather than C∗‑normed, and (iii) potential connections to quantum field theory where Schwartz‑type algebras appear as spaces of test functions. Overall, the work provides a robust homological toolkit that bridges algebraic K‑theory, Kasparov theory, and locally convex analysis, thereby generalizing fundamental results such as Bott periodicity and the Thom isomorphism to a much wider algebraic landscape.