Lifting theorems for completely positive maps

  • James Gabe

    University of Southern Denmark, Odense, Denmark
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We prove lifting theorems for completely positive maps going out of exact CC^*-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if X\mathsf X is a second countable topological space, A\mathfrak A and B\mathfrak B are separable, nuclear CC^*-algebras over X\mathsf X, and the action of X\mathsf X on A\mathfrak A is continuous, then E(X;A,B)KK(X;A,B)E(\mathsf X; \mathfrak A, \mathfrak B) \cong KK(\mathsf X; \mathfrak A, \mathfrak B) naturally. As an application, we show that a separable, nuclear, strongly purely infinite CC^*-algebra A\mathfrak A absorbs a strongly self-absorbing CC^*-algebra D\mathscr D if and only if I\mathfrak I and ID\mathfrak I\otimes \mathscr D are KKKK-equivalent for every two-sided, closed ideal I\mathfrak I in A\mathfrak A. In particular, if A\mathfrak A is separable, nuclear, and strongly purely infinite, then AO2A\mathfrak A \otimes \mathcal O_2 \cong \mathfrak A if and only if every two-sided, closed ideal in A\mathfrak A is KKKK-equivalent to zero.

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James Gabe, Lifting theorems for completely positive maps. J. Noncommut. Geom. 16 (2022), no. 2, pp. 391–421

DOI 10.4171/JNCG/479