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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A separable Brown-Douglas-Fillmore theorem and weak stability

Author: Huaxin Lin
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2889-2925
MSC (2000): Primary 46L05, 46L80
Published electronically: March 2, 2004
MathSciNet review: 2052601
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Abstract: We give a separable Brown-Douglas-Fillmore theorem. Let $A$ be a separable amenable $C^*$-algebra which satisfies the approximate UCT, $B$ be a unital separable amenable purely infinite simple $C^*$-algebra and $h_1, \, h_2: A\to B$ be two monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only if $ [h_1]=[h_2]\,\,\,\,{\rm in}\,\,\, KL(A,B). $ We prove that, for any $\varepsilon>0$ and any finite subset $\mathcal{F}\subset A$, there exist $\delta>0$ and a finite subset $\mathcal{G}\subset A$ satisfying the following: for any amenable purely infinite simple $C^*$-algebra $B$ and for any contractive positive linear map $L: A\to B$ such that

\begin{displaymath}\Vert L(ab)-L(a)L(b)\Vert<\delta\quad{and}\quad \Vert L(a)\Vert\ge (1/2)\Vert a\Vert \end{displaymath}

for all $a\in \mathcal{G},$ there exists a homomorphism $h: A\to B$such that

\begin{displaymath}\Vert h(a)-L(a)\Vert<\varepsilon\,\,\,\,\,{\rm for\,\,\,all}\,\,\, a\in \mathcal{F} \end{displaymath}

provided, in addition, that $K_i(A)$ are finitely generated. We also show that every separable amenable simple $C^*$-algebra $A$ with finitely generated $K$-theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple $C^*$-algebras. As an application, related to perturbations in the rotation $C^*$-algebras studied by U. Haagerup and M. Rørdam, we show that for any irrational number $\theta$ and any $\varepsilon>0$ there is $\delta>0$ such that in any unital amenable purely infinite simple $C^*$-algebra $B$ if

\begin{displaymath}\Vert uv-e^{i\theta\pi}vu\Vert<\delta \end{displaymath}

for a pair of unitaries, then there exists a pair of unitaries $u_1$ and $v_1$ in $B$ such that

\begin{displaymath}u_1v_1=e^{i\theta\pi}v_1u_1,\,\,\,\,\,\Vert u_1-u\Vert<\varepsilon\quad\text{and} \quad\Vert v_1-v\Vert<\varepsilon. \end{displaymath}

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Additional Information

Huaxin Lin
Affiliation: Department of Mathematics, East China Normal University, Shanghai, People’s Republic of China
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Keywords: Weakly semiprojective $C^*$-algebras, purely infinite simple $C^*$-algebras
Received by editor(s): September 18, 2002
Received by editor(s) in revised form: April 29, 2003
Published electronically: March 2, 2004
Additional Notes: This research was partially supported by NSF grant DMS 0097903
Article copyright: © Copyright 2004 American Mathematical Society