A new characterization of (pre)liminary C*-algebras
Martino Lupini
TL;DR
The paper develops a comprehensive ordinal-graded framework for classifying separable C*-algebras via type I$_{α}$ and $α$-subhomogeneous hierarchies, unifying Fell algebras (α=0) and finite-type hierarchies (α∈ω) as special cases. It introduces Fell and Pedersen ranks through weight functions on spectra, proving equivalences: A is type $I_{α}$ iff its Fell rank is at most $1+α$, and A is $α$-subhomogeneous iff its Pedersen rank is at most $α$. The main results establish that every separable liminary algebra sits at some level $I_{α}$ and every algebra with no infinite-dimensional irreducibles sits at some $α$-subhomogeneous level, with strict realizability across all countable ordinals. The work also develops the Fell compactification, weighted spectra, and constructive procedures such as Lazar and Taylor jumps, or tree-based constructions, to realize algebras of arbitrarily high rank. Overall, the paper provides a complete, ordinal-valued stratification of liminary and preliminary C*-algebras, linking descriptive set-theoretic, spectral, and topological methods to operator-algebraic structure.
Abstract
Given an arbitrary countable ordinal $α$, we introduce the notion of type $I_{α}$ C*-algebra and $α$-subhomogeneous C*-algebra. When $α=0$, these recover the notions of Fell C*-algebra and of commutative C*-algebra, respectively. When $α= n <ω$, these recover the notions of type $I_{n}$ C*-algebra and of $n$-subhomogeneous C*-algebra, respectively. We prove that a separable C*-algebra is liminary if and only if it is type $I_{α}$ for some $α<ω_{1}$, and it is preliminary (i.e., has no infinite-dimensional irreducible representation) if and only if it is $α$-subhomogeneous for some $α<ω_{1}$. We also prove that for any countable ordinal $α$ there exists a separable C*-algebra that is type $I_{α}$ and not type $I_{β}$ for $β<α$, and a separable C*-algebra that is $α$-subhomogeneous and not $β$-subhomogeneous for any $β<α$.