Unitary equivalences for essential extensions of 𝐶*-algebras

Lin, Huaxin

doi:10.1090/S0002-9939-09-09921-3

Unitary equivalences for essential extensions of $C^*$-algebras
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Proc. Amer. Math. Soc. 137 (2009), 3407-3420 Request permission

Abstract:

Let $A$ be a unital separable $C^*$-algebra and $B=C\otimes {\mathcal K},$ where $C$ is a unital $C^*$-algebra. Let $\tau : A\to M(B)/B$ be a unital full essential extension of $A$ by $B.$ We show that there is a bijection between elements in a quotient group of $K_0(B)$ onto the strong unitary equivalence classes of unital full essential extensions $\sigma$ for which $[\sigma ]=[\tau ]$ in $KK^1(A, B).$ Consequently, when this group is zero, unitarily equivalent full essential extensions are strongly unitarily equivalent. When $B$ is a non-unital but $\sigma$-unital simple $C^*$-algebra with continuous scale, we also study the problem when two approximately unitarily equivalent essential extensions are strongly approximately unitarily equivalent. A group is used to compute the strongly approximate unitary equivalence classes in the same approximate unitary equivalent class of essential extensions.

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