Spectral ranks and dynamical lengths of preliminary C*-algebras

We classify the spectral ranks and automorphism lengths of nonzero separable preliminary C*-algebras, whose irreducible representations are all finite-dimensional. The spectral invariants are the Fell rank ρ and the Pedersen rank β of the irreducible-dimension function. Automorphism length q measures the transfinite descriptive complexity of automorphisms modulo those implemented by multiplier unitaries; its fractional values distinguish terminal Borel classes.

The possible rank pairs are ρ ∈ {ωα + 1 : α < ω₁} ∪ {ωλ : 0 < λ < ω₁ is limit}, with ρ ≤ β < ω₁. All have realizations with Hausdorff primitive spectrum, and the same Fell-rank list holds for liminary algebras. The possible automorphism lengths are countable ordinals, half-stage values α + ½ for 0 ≤ α < ω₁, and upper fractional values α + ½ + ε for 0 < α < ω₁. The only joint restrictions are that β = 1 forces q = 0, ρ > 1 forces q ≥ 1, and an upper fractional value forces β > ω. Every compatible full frank (ρ, β, q) is realized.

For unital preliminary algebras, derivation and automorphism lengths agree. Their common value is zero exactly when the dimension function is locally constant; then the rank pair is (1, n) for some positive integer n. The positive unital values are precisely the countable successor ordinals and the half-stage and upper fractional values at positive bases. Each occurs on a unital 2-subhomogeneous Fell AF algebra with rank pair (1, ω + 1) and Pedersen length two. This pair is coordinatewise minimal for positive unital dynamics. In contrast, nonunital degree-two homogeneous algebras realize every ordinal and half-stage automorphism length already at rank pair (1, 2).

GitHub repository: https://github.com/eulogica/spectral-ranks-automorphism-complexity