Shape theory and extensions of C*-algebras

Abstract

Let $A$, $A^{\prime }$ be separable $C^{\ast }$-algebras, $B$ a stable $\sigma$-unital $C^{\ast }$-algebra. Our main result is the construction of the pairing $[[A^{\prime },A]]\times {\mathrm{Ext}}^{-1/2}(A,B)\to {\mathrm{Ext}}^{-1/2}(A^{\prime },B)$, where $[[A^{\prime },A]]$ denotes the set of homotopy classes of asymptotic homomorphisms from $A^{\prime }$ to $A$ and ${\mathrm{Ext}}^{-1/2}(A,B)$ is the group of semi-invertible extensions of $A$ by $B$. Assume that all extensions of $A$ by $B$ are semi-invertible. Then this pairing allows us to give a condition on $A^{\prime }$ that provides semi-invertibility of all extensions of $A^{\prime }$ by $B$. This holds, in particular, if $A$ and $A^{\prime }$ are shape equivalent. A similar condition implies that if ${\mathrm{Ext}}^{-1/2}$ coincides with $E$-theory (via the Connes-Higson map) for $A$ then the same holds for $A^{\prime }$.

23 pages

References in corpus (3)

• A new application of Random Matrices: Ext(C*_{red}(F_2)) is not a group
• Some Examples of Blackadar and Kirchberg's MF Algebras
• On the lack of inverses to C*-extensions related to property T groups

Cited by in corpus (1)

• Halmos' two projections theorem for Hilbert $C^{\ast }$-module operators and the Friedrichs