The Ext--Group of Unitary Equivalence Classes of Unital Extensions

Abstract

Let $\A$ be a unital separable nuclear $C^{\ast }$--algebra which belongs to the bootstrap category $\mathbb{N}$ and $\B$ be a separable stable $C^{\ast }$--algebra. In this paper, we consider the group $\Ext_u(\A,\B)$ consisting of the unitary equivalence classes of unital extensions $τ\colon\A\rightarrow Q(\B)$. The relation between $\Ext_u(\A,\B)$ and $\Ext(\A,\B)$ is established. Using this relation, we show the half--exactness of $\Ext_u(\cdot,\B)$ and the (UCT) for $\Ext_u(\A,\B)$. Furthermore, under certain conditions, we obtain the half--exactness and Bott periodicity of $\Ext_u(\A,\cdot)$.

arXiv comment16 pages Acta Math. Sinica (English series)(to appear)