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A separable Brown-Douglas-Fillmore theorem and weak stability

Authors:	Huaxin Lin

Institution:	Department of Mathematics, East China Normal University, Shanghai, People's Republic of China

Abstract:	We give a separable Brown-Douglas-Fillmore theorem. Let be a separable amenable -algebra which satisfies the approximate UCT, be a unital separable amenable purely infinite simple -algebra and $h_1, \, h_2: A\to B$ be two monomorphisms. We show that and 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 -algebra 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 are finitely generated. We also show that every separable amenable simple -algebra with finitely generated -theory which is in the so-called bootstrap class is weakly stable with respect to the class of amenable purely infinite simple -algebras. As an application, related to perturbations in the rotation -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 -algebra 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 and in such that $\begin{displaymath}u_1v_1=e^{i\theta\pi}v_1u_1,\,\,\,\,\,\Vert u_1-u\Vert<\varepsilon\quad\text{and} \quad\Vert v_1-v\Vert<\varepsilon. \end{displaymath}$

Keywords:	Weakly semiprojective $C^$-algebras purely infinite simple $C^$-algebras

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