Dynamical spectra of unital C*-algebras: AF realizations and Cartan pairs
Description
For a separable unital C*-algebra, the dynamical length records the descriptive complexity of its inner derivations and inner automorphisms. For a Cartan pair, it also records two cohomological lengths of the Weyl groupoid and the length of the automorphism group preserving the Cartan subalgebra. We determine the spectra of AF algebras, standard AF pairs, C*-diagonals in AF algebras, and arbitrary Cartan pairs whose ambient algebra is AF. Nonconstant tuples form a finite list at the bottom of the scale. The positive constant coordinates are exactly {β+1 : β<ω₁} ∪ {α+1/2, α+1/2+ε : 0<α<ω₁}, where the half-step symbols are formal indices in the scale. We construct the fractional realizations through zero-dimensional incidence systems and prove that every positive derivation length of a separable unital C*-algebra has a standard AF-pair realization. Each positive value has a standard diagonal realization in a separable unital 2-subhomogeneous AF algebra satisfying Fell's condition. The absolute spectrum of all separable unital C*-algebras coincides with the AF spectrum, and the spectrum of Kumjian diagonal pairs coincides with that of standard AF pairs. General Cartan pairs have one additional low value. We also determine the spectra within the scattered, continuous-trace, simple, and homogeneous classes. The exclusion of the remaining fractional candidates is imported from a companion theorem.