Recursive classification and admissible descent for preliminary C*-algebras with prescribed weighted spectrum

We prove that compatible local separable preliminary C*-algebras on an open cover occur as the specified ideals of a global algebra exactly when their transported corona ranges are pairwise orthogonal. Here preliminary means that all irreducible representations are finite-dimensional. We fix a primitive spectrum X and positive integer representation dimensions w, with X second countable, locally compact, and sober and w satisfying the dimension-compatibility condition for preliminary spaces; realization of (X,w) is not assumed. The canonical homogeneous filtration is countable. Classical homogeneous classification and Busby extension theory then give an exact recursive classification by matrix bundles and attaching maps subject to a localized-kernel test for the prescribed topology. We organize these extension data by Čech cocycles satisfying corona orthogonality, modulo gauge and refinement, without reducing them to characteristic classes. A doubled-endpoint example exhibits a nonrealizable class in unrestricted nonabelian H¹. For constant weight n, admissibility is automatic and the invariant is the marked classical Fell–Tomiyama–Takesaki invariant Čech H¹(X,PU(n)). Geometric boundary models for two-stratum algebras allow singular multiplier fibres; explicit families yield attachment multiplicities and Fell ranks.

GitHub repository: eulogica/preliminary-classification