# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2018mar18.2223

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

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