×

zbMATH — the first resource for mathematics

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)
PDF BibTeX XML Cite
Full Text: DOI Link