On the classification of simple inductive limit $C^*$-algebras. I: The reduction theorem

By G. Gong. Documenta Mathematica, 2002

122 citations recorded by the source corpus.

Suppose that formula math. is a simple C*-algebra, where X n,i are compact metrizable spaces of uniformly bounded dimensions (this restriction can be relaxed to a condition of very slow dimension growth). It is proved in this article that A can be written as an inductive limit of direct sums of matrix algebras over certain special 3-dimensional spaces. As a consequence it is shown that this class of inductive limit C*-algebras is classified by the Elliott invariant - consisting of the ordered K-group and the tracial state space - in a subsequent paper joint with G. Elliott and L. Li (Part II of this series). (Note that the C*-algebras in this class do not enjoy the real rank zero property.).

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f85c5045529b98ac5fd4602d217416b7a0002b2

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Citations in the corpus

Finite cyclic group actions with the tracial Rokhlin property (2006, methodology): “1, the classification theorem of [7] does not apply to the direct limit, and we do not know of a classification proof which does not depend on the tracial Rokhlin property.”
Regularity properties in the classification program for separable amenable C*-algebras (2007, unknown): “4 (Elliott-Gong-Li and Gong, [23] and [26]).”
Completeness of the isomorphism problem for separable C*-algebras (2013, background): “g certain classes of real rank zero algebras, AH algebras of slow dimension growth or separable, simple, purely infinite, nuclear algebras (modulo the the universal coefficients theorem) [18, 21, 22, 46, 23, 63, 47] and there have been dramatic breakthroughs in the program, including the counterexam”
The Elliott conjecture for Villadsen algebras of the first type (2006, unknown): “We will prove that A has very slow dimension growth in the sense of Gong; bounded dimension growth follows by the reduction theorem of [15] or, alternatively, the classification theorem of [20].”
The Tracial Topological Rank of C*‐Algebras (2001, background): “It also absorbs material from [15, 13] as well as many other earlier works like [11, 7, 4, 30], to name a few.”