Classification of simple C*-algebras of tracial topological rank zero

By Huaxin Lin. 2000

193 citations recorded by the source corpus.

We give a classification theorem for unital separable simple nuclear $C^*$-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if $A$ and $B$ are two such $C^*$-algebras and $$ (K_0(A), K_0(A)_+, [1_A], K_1(A)) $$ $$ \cong (K_0(B), K_0(B)_+, [1_B], K_1(B)), $$ then $A\cong B.$

Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6a363a96920ddf031143b9751754d496f37eb173

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Citations in the corpus

Dimension, comparison, and almost finiteness (2017, background): “The notion of tracial rank, which has its roots in work of Gong [9] and Popa [35] and was formalized and applied by Lin in his seminal work of the 1990s as a way to circumvent inductive limit hypotheses in the stably finite case [24, 25, 26], continues to do much of the technical legwork in classifi”
Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras (2008, background): “It follows from [24] that C is a unital simple AH-algebra.”
A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, methodology): “The costruction is similar to that of Section 2 of [58], and roughly, we will construct a map factors through the C∗-subalgebra (in C0) of the given C∗-algebra in B0, and also require this map to carry the given KL-element.”
Regularity properties in the classification program for separable amenable C*-algebras (2007, background): “Lin proved that simple unital separable amenable C-algebras of tracial rank zero satisfy the Elliott conjecture, modulo the ever present UCT assumption ([38]).”
Approximate Homotopy of Homomorphisms from C(x) Into a Simple C*-Algebra (2006, background): “We refer the reader to [24], [28], [25], [27], [3], [45] and [46] for more information.”
COVERING DIMENSION AND QUASIDIAGONALITY (2002, background): “In subsequent work the following partial result will be proved: If A is separable, simple and unital with real rank zero, stable rank one, finite decomposition rank, weakly unperforated K0-group and a unique tracial state, then A has tracial rank zero in the sense of [11].”
Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras (2014, methodology): “Let us proceed as that of Section 2 of [53].”
The tracial Rokhlin property for actions of finite groups on C*-algebras (2006, background): “The classification [17] of simple separable nuclear C*-algebras with tracial rank zero and satisfying the Universal Coefficient Theorem can be regarded as a vast generalization of the classification of AF algebras.”
Subalgebras of simple AF-algebras (2018, methodology): “Also, the classification results for simple, nuclear C∗-algebras with tracial approximation structure, initiated by Lin in [50] and culminating in the remarkable work of Gong, Lin, and Niu in [35], depends heavily on classification results for embeddings between such algebras.”
Quasidiagonality of nuclear C*-algebras (2015, unknown): “1”
Crossed Products of the Cantor Set by Free Minimal Actions of ℤd (2002, result): “Therefore, by recent work of H. Lin [ 27 ], a positive”