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Full extensions and approximate unitary equivalence. (English) Zbl 1152.46049
Summary: Let $$A$$ be a unital separable amenable $$C^\ast$$-algebra and let $$C$$ be a unital $$C^\ast$$-algebra with a certain infinite property. We show that two full monomorphisms $$h_1,h_2 : A \rightarrow C$$ are approximately unitarily equivalent if and only if $$[h_1] = [h_2]$$ in $$KL(A,C)$$. Let $$B$$ be a nonunital but $$\sigma$$-unital $$C^\ast$$-algebra for which $$M(B) / B$$ has a certain infinite property. We prove that two full essential extensions are approximately unitarily equivalent if and only if they induce the same element in $$KL(A,M(B) / B)$$. The set of approximately unitarily equivalence classes of full essential extensions forms a group. If $$A$$ satisfies the universal coefficient theorem, the group can be identified with $$KL(A,M(B) / B)$$.

##### MSC:
 46L05 General theory of $$C^*$$-algebras 46L35 Classifications of $$C^*$$-algebras 46L80 $$K$$-theory and operator algebras (including cyclic theory)
##### Keywords:
extension of $$C^*$$-algebras; simple $$C^*$$-algebras
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