Dynamical length of Cartan pairs: comparison, rigidity, and invariance

A Cartan subalgebra permits a separable unital C*-algebra to be studied through its twisted étale Weyl groupoid. We combine five cohomological, derivation, and automorphism lengths into the dynamical length ℓ(A,D) of a Cartan pair. These lengths measure the accumulation of coboundaries and inner symmetries through the canonical filtration of phantom subgroups.

We determine the universal order and equality relations among the five coordinates. Exactly six dynamical lengths occur when the circle-valued cohomological length is at most one; at and above one, all five coordinates agree. Four coordinates depend only on the Weyl groupoid. The fifth can depend on the twist, even for a compact amenable groupoid, and this dependence is confined to one low-complexity case. Residual topological principality alone ensures twist independence, via a geometric decomposition at vanishing bounded cohomology. We derive rigidity criteria and prove invariance under continuous categorical equivalence and, for ample Weyl groupoids, Morita–Kakutani equivalence, with the same possible exception for the full invariant.

A compact-incidence obstruction excludes σ + ½ − ε for every countable successor ordinal σ from the absolute derivation and automorphism lengths of every separable unital C*-algebra, and hence from all five Cartan coordinates. Finite direct sums satisfy a coordinatewise maximum formula. For locally compact Polish groupoids with compact units, we exclude (3/2, 3/2 − ε) and realize (1, ½ + ε) by a principal groupoid with nonopen source.

Manuscript repository: GitHub repository.