Conformal transformations and equivariance in unbounded KK-theory

We extend unbounded Kasparov theory to encompass conformal group and quantum group equivariance. This new framework allows us to treat conformal actions on both manifolds and noncommutative spaces. As examples, we present unbounded representatives of Kasparov’s $γ$-element for the real and complex Lorentz groups and display the conformal $SL_q(2)$-equivariance of the standard spectral triple of the Podleś sphere. In pursuing descent for conformally equivariant cycles, we are led to a new framework for representing Kasparov classes. Our new representatives are unbounded, possess a dynamical quality, and also include known twisted spectral triples. We define an equivalence relation on these new representatives whose classes form an abelian group surjecting onto KK. The technical innovation which underpins these results is a novel multiplicative perturbation theory. By these means, we obtain Kasparov classes from the bounded transform with minimal side conditions.

💡 Research Summary

The paper develops a comprehensive extension of unbounded Kasparov theory that incorporates conformal group actions and quantum group equivariance. Traditional unbounded Kasparov modules consist of a C∗‑algebra A, a Hilbert B‑module E, and a regular self‑adjoint operator D. The authors introduce “conformal transformations” between such cycles as pairs (U, µ) where U is a unitary between the underlying modules and µ is a bounded, invertible “conformal factor”. The key requirement is that for a dense subalgebra of A the difference U∗D₂U − µD₁µ∗ is bounded. This notion generalizes Kucerovsky’s uniform equivariance, which only captures isometric or nearly isometric actions, and allows genuine conformal (non‑isometric) symmetries.

The technical heart of the work is a novel multiplicative perturbation theory. While additive perturbations (D → D+A) are classical, the authors study perturbations of the form D → µDµ∗. They prove that, under suitable boundedness and domain conditions, any perturbation that preserves the KK‑class of the bounded transform must be expressible as µDµ∗ + A. Moreover, they introduce a logarithmic transform L(D)=F(D)·log((1+D²)^{1/2}) (with F(D) the usual bounded transform) which converts multiplicative perturbations into additive ones, thereby allowing the use of standard additive techniques.

Using these tools, the paper defines several levels of equivariance:

1. Uniform equivariance (the classical Kucerovsky setting).
2. Conformal group equivariance for ordinary locally compact groups.
3. Uniform quantum‑group equivariance for Hopf‑C∗‑algebras.
4. Conformal quantum‑group equivariance.

For each, the authors construct descent maps and dual Green–Julg maps, showing that the bounded transform of a conformally equivariant unbounded cycle yields an equivariant bounded Kasparov module, and that the logarithmic transform produces a uniformly equivariant unbounded cycle. This machinery is applied to several important examples:

• γ‑elements for the real Lorentz groups SO(2n+1,1), SO(2n,1) and the complex Lorentz group SU(n,1). The authors give explicit unbounded representatives of these elements, which were previously known only in bounded form.
• A second‑order spectral triple for the C∗‑algebra of the Heisenberg group, equivariant under the dilation action, illustrating a genuinely noncommutative conformal example.
• Conformal quantum‑group equivariance of the standard spectral triple on the Podleś sphere under the action of the quantum group SL_q(2). This recovers known twisted spectral triples without invoking Lipschitz regularity.

The paper then introduces the concept of conformally generated cycles. These are unbounded cycles equipped with a conformal factor and a “matched operator” (an operator locally bounded over the right‑hand algebra) that together encode both geometric and dynamical data. All known twisted spectral triples fit into this framework, and the authors show that their bounded transforms are well‑defined without extra smoothness assumptions.

To obtain a homology‑type theory, the authors generalize Cuntz–Skandalis cobordism to unbounded cycles. They prove that cobordism classes form a ℤ/2‑graded abelian group surjecting onto KK. Further, they define an equivalence relation called conformism on conformally generated cycles, essentially the cobordism generated by conformal and singular conformal transformations. The resulting conformism classes also form an abelian group surjecting onto KK.

Technical appendices supply the necessary functional‑analytic background: fractional powers of positive regular operators on Hilbert modules, Hilbert modules over locally compact spaces, the theory of matched operators (a pro‑C∗‑algebra of regular operators), and a characterization of compactly supported states via the Pedersen ideal.

In summary, the authors provide a robust multiplicative perturbation theory that bridges conformal and quantum‑group symmetries with unbounded KK‑theory. Their framework yields new unbounded representatives for important KK‑elements, unifies twisted spectral triples under a single “conformally generated” umbrella, and establishes a refined equivalence relation that recovers the usual KK‑group while retaining richer geometric information. This work opens avenues for applying KK‑theory to conformally invariant physical models, non‑isometric group actions, and quantum‑group symmetries in noncommutative geometry.