Non-amenable simple C*-algebras with tracial approximation

2021/01/19 by Xuanlong Fu, Fu, Xuanlong, Huaxin Lin +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.2101.07900

openalex publication_date 2021/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct two types of unital separable simple C^*-alebras A_z^C₁ and A_z^C₂, one is exact but not amenable, and the other is non-exact. Both have the same Elliott invariant as the Jiang-Su algebra, namely, A_z^C_i has a unique tracial state, (K₀(A_z^C_i), K₀(A_z^C_i)₊, [1_Az^Ci ])=(\mathbb Z, \mathbb Z₊,1) and K₁(A_z^C_i)=\0\ (i=1,2). We show that A_z^C_i (i=1,2) is essentially tracially in the class of separable \cal Z-stable C^*-alebras of nuclear dimension 1. A_z^C_i has stable rank one, strict comparison for positive elements and no 2-quasitrace other than the unique tracial state. We also produce models of unital separable simple non-exact C^*-alebras which are essentially tracially in the class of simple separable nuclear \cal Z-stable C^*-alebras and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.

Citations

• Simple stably projectionless C*-algebras with generalized tracial rank one

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• Crossed products by compact group actions with the weak tracial Rokhlin property

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