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

Given an arbitrary countable ordinal $\alpha $, we introduce the notion of type $I_{\alpha }$ C*-algebra and $\alpha $-subhomogeneous C*-algebra. When $\alpha =0$, these recover the notions of Fell C*-algebra and of commutative C*-algebra, respectively. When $\alpha =n<\omega $, 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 $I_{\alpha }$ for some $\alpha<\omega _{1}$, and it is preliminary (i.e., has no infinite-dimensional irreducible representation) if and only if it is $\alpha $-subhomogeneous for some $\alpha<\omega _{1}$. We also prove that for any countable ordinal $\alpha $ there exists a separable C*-algebra that is type $I_{\alpha }$ and not type $I_{\beta }$ for $\beta<\alpha $, and a separable C*-algebra that is $\alpha $-subhomogeneous and not $\beta $-subhomogeneous for any $\beta<\alpha $.