Correctness of the uniform equivariance definition for unbounded KK-theory

Determine whether the uniform G-equivariant order-(1/(1−α)) A–B cycle definition given in Definition \ref{definition:ordinary-unbounded-equivariance} is the correct generalisation of equivariance to unbounded KK-theory. Ascertain its adequacy across geometrically relevant cases or identify precise conditions under which it fails and what refinement is required.

Background

The authors extend Kucerovsky’s framework to higher-order unbounded Kasparov modules and introduce a notion of uniform group equivariance (Definition \ref{definition:ordinary-unbounded-equivariance}). They note discrepancies between equivariance in the bounded picture and what can be captured by standard unbounded formulations, particularly when conformal actions are present (e.g., conformal Dirac examples show bounded transforms are equivariant under larger groups than the unbounded Dirac operators).

To address these limitations, the paper develops a conformal equivariance framework and associated technical tools, but it explicitly questions whether the original uniform equivariance definition truly generalises equivariance for unbounded KK-theory. This leaves open the task of validating or refining that definition in full generality.