On the Decomposition Theorems for C *-Algebras
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Chinese Annals of Mathematics, Series B c The Editorial Office of CAM and
Springer-Verlag Berlin Heidelberg 2020
On the Decomposition Theorems for C ∗-Algebras Chunlan JIANG1
Liangqing LI2
Kun WANG3
Abstract Elliott dimension drop interval algebra is an important class among all C ∗ algebras in the classification theory. Especially, they are building stones of AHD algebra and the latter contains all AH algebras with the ideal property of no dimension growth. In this paper, the authors will show two decomposition theorems related to the Elliott dimension drop interval algebra. Their results are key steps in classifying all AH algebras with the ideal property of no dimension growth. Keywords C ∗ -algebra, Elliott dimension drop interval algebra, Decomposition theorem, Spectral distribution property 2000 MR Subject Classification 46L35, 46L80
1 Introduction Classification theorems have been obtained for AH algebras—the inductive limits of cut downs of matrix algebras over compact metric spaces by projections—and AD algebras—the inductive limits of Elliott dimension drop interval algebras in two special cases: (1) Real rank zero case: All such AH algebras with no dimension growth and such AD algebras (see [1–4, 7–8, 12–17]). (2) Simple case: All such AH algebras with no dimension growth (which includes all simple AD algebras by [9]) (see [5–6, 11, 18, 27–32, 39–40]). In [11], the authors pointed out two important possible next steps after the completion of classification of simple AH algebras (with no dimension growth). One of these is the classification of simple ASH algebras—the simple inductive limits of subhomogeneous algebras (with no dimension growth). The other is to generalize and unify the above-mentioned classification theorems for simple AH algebras and real rank zero AH algebras by classifying AH algebras with the ideal property. The ideal properties in the classification theory are intensively studied previously (see [35–36, 41–42])In particular, ASI and AI algebras with the ideal property are classified by the Stevens-Jiang invariant (see [22, 26, 41])In this article, we have achieved several key results for the second goal by providing two decomposition theorems. As in [8], let TII,k be the 2-dimensional connected simplicial complex with H 1 (TII,k ) = 0 and H 2 (TII,k ) = Z/kZ, and let Ik be the subalgebra of Mk (C[0, 1]) defined by Ik = {f ∈ Mk (C[0, 1]) : f (0) ∈ C · 1k and f (1) ∈ C · 1k }. Manuscript received May 18, 2019. of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China. E-mail: [email protected] 2 Department of Mathematics, University of Puerto Rico at Rio Piedras, PR 00936, USA. E-mail: [email protected] 3 Corresponding author. Department of Mathematics, University of Puerto Rico at Rio Piedras, PR 00936, USA. E-mail: [email protected] 1 College
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C. L. Jiang, L. Q. Li and K. Wang
This algebra is called an Elliott dimension drop interval algebra. Denote by HD the class of algebras consisting of direct sums of building blocks of the forms M
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