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 A, using the ideal I At (A) generated by the minimal projections of 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 Aα, along an uncountable limit ordinal λ, such that A α+1 /Aα is * -isomorphic to the algebra of all compact operators on a separable Hilbert space and A α+1 is σ-unital and stable for each α < λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C * -algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rørdam in the nonseparable case. All the above C * -algebras Aαs 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 [25,26].Proof. We prove J α = I α ⊗ K(ℓ 2 ) by induction on α ≤ ht(A). For α = 1 this follows from Lemma 5.2. At a successor ordinal by Lemma 5.2 and the inductive assumption we have J α+1 /J α =

A nonstable 𝐶*-algebra with an elementary essential composition series

“…A partial version of Theorem 1.4 for an inductive limit of length possibly bigger than the first uncountable ordinal ω 1 of possibly nonseparable C * -algebras was obtained in Theorem 7.7. of [11].…”

“…(3) An extension of a separable stable AF-algebra by a separable stable AFalgebra is stable (Blackadar [3], cf. 6.12 of [28], 7.3 of [11]). (4) Countable inductive limits of separable stable algebras are stable (Hjelmborg and Rørdam 4.1 of [16]).…”

“…Some of many equivalent conditions defining scattered C * -algebras are: every nonzero quotient has a minimal projection or the spectrum of every self-adjoint element is at most countable ( [33]) or every subalgebra is LF [19]. For a recent survey on scattered C * -algebras see [11]. Commutative scattered C * -algebras are exactly of the form C 0 (X) where X is locally compact and scattered, i.e., its every nonempty (closed) subset has a relatively isolated point.…”

“…In fact, such spaces must be totally disconnected and so, by the Stone duality they correspond exactly to superatomic Boolean algebras ( [24]). In [11] we introduced a canonical composition series (I α (A)) α≤ht(A) for a scattered C * -algebra A which corresponds to the Cantor-Bendixson derivatives in the commutative case. I α+1 (A) is defined by requiring that I α+1 (A)/I α (A) is the subalgebra generated in A/I α (A) by all minimal projections.…”

“…It turns out that being fully noncommutative is equivalent to a strong noncommutativity condition, namely that the center of the multiplier algebra of any quotient is trivial (Proposition 6.3 of [11]). Also for a fully noncommutative scattered C * -algebra the Cantor-Bendixson composition series coincides with the GCR composition series (Proposition 6.4 of [11]) and full noncommutativity is equivalent to the stability for separable scattered C * -algebras such that κ α = ω for all α < ht(A) (Lemma 7.3 of [11]). So our construction yields the following: Theorem 1.7.…”