TY - JOUR

T1 - Lifting theorems for completely positive maps

AU - Gabe, James

N1 - Funding Information:
Funding. This work was supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
Publisher Copyright:
© 2022 European Mathematical Society Published by EMS Press.

PY - 2022/9/11

Y1 - 2022/9/11

N2 - We prove lifting theorems for completely positive maps going out of exact C*-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if X is a second countable topological space, A and B are separable, nuclear C -algebras over X, and the action of X on A is continuous, then E.XI A; B/ Š KK.XI A; B/ naturally. As an application, we show that a separable, nuclear, strongly purely infinite C* -algebra A absorbs a strongly self-absorbing C* -algebra D if and only if I and I ⊗ D are KK-equivalent for every two-sided, closed ideal I in A. In particular, if A is separable, nuclear, and strongly purely infinite, then A ⊗ O2 Š A if and only if every two-sided, closed ideal in A is KK-equivalent to zero.

AB - We prove lifting theorems for completely positive maps going out of exact C*-algebras, where we remain in control of which ideals are mapped into which. A consequence is, that if X is a second countable topological space, A and B are separable, nuclear C -algebras over X, and the action of X on A is continuous, then E.XI A; B/ Š KK.XI A; B/ naturally. As an application, we show that a separable, nuclear, strongly purely infinite C* -algebra A absorbs a strongly self-absorbing C* -algebra D if and only if I and I ⊗ D are KK-equivalent for every two-sided, closed ideal I in A. In particular, if A is separable, nuclear, and strongly purely infinite, then A ⊗ O2 Š A if and only if every two-sided, closed ideal in A is KK-equivalent to zero.

KW - ideal related KK-theory

KW - Lifting completely positive maps

KW - strongly self-absorbing C -algebras

U2 - 10.4171/JNCG/479

DO - 10.4171/JNCG/479

M3 - Journal article

AN - SCOPUS:85139603962

SN - 1661-6952

VL - 16

SP - 391

EP - 421

JO - Journal of Noncommutative Geometry

JF - Journal of Noncommutative Geometry

IS - 2

ER -