A new characterization of (pre)liminary C*-algebras

2025/09/25 by Lupini, Martino
#03E15 #46L40 #46L57 (Primary) #54H05 (Secondary) #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Logic (math.LO) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2509.20980

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<ω₁, and it is preliminary (i.e., has no infinite-dimensional irreducible representation) if and only if it is α-subhomogeneous for some α<ω₁. 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.

Citations

• Prime ideals in C*-algebras and applications to Lie theory
• Commutativity of central sequence algebras
• The generator rank of subhomogeneous C*-algebras
• Elements of C^*-algebras Attaining Their Norm in a Finite-Dimensional Representation
• A stratified homotopy hypothesis
• Models for subhomogeneous C*-algebras
• The equivalence relations of local homeomorphisms and Fell algebras
• Quotient Spaces Determined by Algebras of Continuous Functions
• The transformation groups whose C^*-algebras are Fell algebras

Related

• Recursive subhomogeneous algebras
• On the classification of simple approximately subhomogeneous C*-algebras not necessarily of real rank zero
• Subalgebras of finite codimension in semiprojective C*-algebras
• On the classification problem for C*-algebras
• A Note On Subhomogeneous C*-Algebras