Bivariant compact quantum metric spaces

Develop a correct and generally accepted notion of a bivariant compact quantum metric space suitable for capturing the metric control needed in functorial constructions and products, beyond existing attempts such as the framework proposed in Austad–Kaad–Kyed (2025).

Background

The authors articulate a guiding principle for lifting metric data via unbounded KK-theory, but note a conceptual gap: a precise bivariant metric framework that would formalize the needed "sufficient control" remains to be formulated.

A recent attempt has been made in Austad–Kaad–Kyed (2025), but the paper emphasizes that a definitive notion is still lacking.