K\"unneth Splittings and Classification of C*-Algebras with Finitely Many Ideals
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Publication:4356033
zbMATH Open0880.46048arXiv2005.10475MaRDI QIDQ4356033
Publication date: 8 February 1998
Abstract: The class of AD algebras of real rank zero is classified by an exact sequence of K-groups with coefficients, equipped with certain order structures. Such a sequence is always split, and one may ask why, then, the middle group is relevant for classification. The answer is that the splitting map can not always be chosen to respect the order structures involved. This may be rephrased in terms of the ideals of the C*-algebras in question. We prove that when the C*-algebra has only finitely many ideals, a splitting map respecting these always exists. Hence AD algebras of real rank zero with finitely many ideals are classified by (classical) ordered K-theory. We also indicate how the methods generalize to the full class of ASH algebras with slow dimension growth and real rank zero.
Full work available at URL: https://arxiv.org/abs/2005.10475
ASH algebrasorder structuresdimension growthordered \(K\)-theorysplitting mapAD algebras of real rank zeroexact sequence of \(K\)-groupsideals of \(c^*\)-algebras
(K)-theory and operator algebras (including cyclic theory) (46L80) Classifications of (C^*)-algebras (46L35) (K_0) as an ordered group, traces (19K14)
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