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 $\beta <\alpha$.