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

2016/11/01 by Ghasemi, Saeed, Koszmider, Piotr · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Logic (math.LO) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1611.00221

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_α, along an uncountable limit ordinal λ, such that \mathcal A_α+1/\mathcal A_α is *-isomorphic to the algebra of all compact operators on a separable Hilbert space and \mathcal A_α+1 is σ-unital and stable for each α

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