Noncommutative Cantor-Bendixson derivatives and scattered $C^*$-algebras

Abstract

We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered $C^*$-algebra $\mathcal A$, using the ideal $\mathcal I^{At}(\mathcal A)$ generated by the minimal projections of $\mathcal A$. With its help, we present some fundamental results concerning scattered $C^*$-algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered $C^*$-algebras and new open problems. In particular, we construct a type $I$ $C^*$-algebra which is the inductive limit of stable ideals $\mathcal A_\alpha$, along an uncountable limit ordinal $\lambda$, such that $\mathcal A_{\alpha+1}/\mathcal A_\alpha$ is $*$-isomorphic to the algebra of all compact operators on a separable Hilbert space and $\mathcal A_{\alpha+1}$ is $\sigma$-unital and stable for each $\alpha<\lambda$, but $\mathcal A$ is not stable and where all ideals of $\mathcal A$ are of the form $\mathcal A_\alpha$. In particular, $\mathcal A$ is a nonseparable $C^*$-algebra with no ideal which is maximal among the stable ideals. This answers a question of M. R\ordam in the nonseparable case. All the above $C^*$-algebras $A_\alpha$s and $A$ satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers.