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

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 $β<α$.