On unital absorbing extensions of C${}^{\ast }$-algebras of stable rank one and real rank zero

Abstract

Suppose that $A,B$ are nuclear, separable ${\mathrm{C}}^{\ast }$-algebras of stable rank one and real rank zero, $A$ is unital simple, $B$ is stable and $({\mathrm{K}}_0(B),{\mathrm{K}}_0^{+}(B))$ is weakly unperforated in the sense of Elliott \cite{Ell}. We show that any unital extension with trivial index maps of $A$ by $B$ is absorbing.