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Journal of Operator Theory

Volume 81, Issue 2, Spring 2019  pp. 481-498.

Functorial properties of $\mathrm{Ext}_\mathrm u(\cdot , \mathcal{B})$ when $\mathcal{B}$ is simple with continuous scale

Authors:  P.W. Ng (1), Tracy Robin (2)
Author institution:(1) Department of Mathematics, Univ. of Louisiana at Lafayette, P. O. Box 43568, Lafayette, LA, 70504-3568, U.S.A.
(2) Department of Mathematics, Prairie View A & M University, P. O. Box 519 -- Mailstop 2225, Prairie View, TX, 77446-0519, U.S.A.

Summary: In this note we define two functors $\mathrm{Ext}$ and $\mathrm{Ext}_\mathrm u$ which capture unitary equivalence classes of extensions in a manner which is finer than $KK^1$. We prove that for every separable nuclear $C^*$-algebra $\mathcal{A}$, and for every $\sigma$-unital nonunital simple continuous scale $C^*$-algebra $\mathcal{B}$, $\mathrm{Ext}(\mathcal{A}, \mathcal{B})$ is an abelian group. We have a similar result for $\mathrm{Ext}_\mathrm u$. We study some functorial properties of the covariant functor $X \mapsto \mathrm{Ext}_\mathrm u(C(X), \mathcal{B})$, where $X$ ranges over the category of compact metric spaces.

Keywords: K-theory, extension theory, Brown--Douglas--Fillmore theory, real rank zero

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