Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras

By Huaxin Lin. 2008

98 citations recorded by the source corpus.

Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and suppose that $\phi_1, \phi_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*\phi_1(c)u_t=\phi_2(c)\tforal c\in C $$ if and only if $[\phi_1]=[\phi_2]$ in $KK(C,A),$ $\phi_1^{\ddag}=\phi_2^{\ddag},$ $(\phi_1)_T=(\phi_2)_T$ and a rotation related map $\bar{R}_{\phi_1,\phi_2}$ associated with $\phi_1$ and $\phi_2$ is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\cal A}$ of unital separable simple amenable \CA s which is strictly larger than the class of separable \CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One consequence of the main result is that all unital simple AH-algebras which are ${\cal Z}$-stable are isomorphic to ones with no dimension growth.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dd973a984328e6ea835f003d4d526ec081ccbe0f

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In Algebra and Number Theory

1. Z. Niu
2. G. Gong
3. Huaxin Lin
4. W. Winter
5. N. Christopher Phillips
6. Andrew S. Toms
7. G. Elliott
8. David Kerr

In Mathematical Physics

1. Z. Niu
2. G. Gong
3. Huaxin Lin
4. W. Winter
5. Andrew S. Toms
6. N. Christopher Phillips
7. G. Elliott
8. David Kerr

Citations in the corpus

• A classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one. (2018, background): “We were then able in [68] to extend this result to the class A, the class of all unital simple separable amenable Z-stable C∗-algebras which satisfy the UCT and have finite tracial rank after tensoring some infinite dimensional UHF-algebra.”
• A classification of finite simple amenable Z-stable C*-algebras, II, --C*-algebras with rational generalized tracial rank one (2019, background): “5 of [40].”
• Classification of finite simple amenable ${\cal Z}$-stable $C^*$-algebras (2014, background): “15 of [63] for Cn (in place of A) and Pn (in place of P).”