Noncommutative probe algebras and classification phenomena

Construct a natural observable that is NIP but unstable with a noncommutative probe algebra, locate it in the Todorčević classification, and determine whether noncommutativity produces classification phenomena unavailable to observables with commutative probe algebras, such as different minimal families arising from a given dense sequence of preparations.

Background

Most physical examples in the paper have commutative probe algebras, although the general framework applies to arbitrary observables. The authors therefore ask whether a genuinely noncommutative probe algebra can support an NIP but unstable observable and whether noncommutativity changes which canonical minimal families can arise from a fixed dense sequence of preparations.

References

Can noncommutativity produce a phenomenon in the classification that observables with commutative probe algebras cannot, for instance in which minimal families of the heptachotomy arise from a given dense sequence of preparations?

— Classification of bipartite observables II: NIP observables and the Todorcevic trichotomy  (2610.12908 - Dueñez et al., 9 Oct 2026) in Section Questions, item 3; see also Remark 2.14

Can the Todor\v{c}evi\ c trichotomy, or its consequences for observables, be proved without forcing?

— Classification of bipartite observables II: NIP observables and the Todorcevic trichotomy  (2610.12908 - Dueñez et al., 9 Oct 2026) in Section Questions, item 10; discussion following Theorem 2.8