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

Let $\text{A}$ be a unital separable nuclear $C^{\ast }$–algebra which belongs to the bootstrap category $\mathbb{N}$ and $\text{B}$ be a separable stable $C^{\ast }$–algebra. In this paper, we consider the group ${\text{Ext}}_u(\text{A},\text{B})$ consisting of the unitary equivalence classes of unital extensions $\tau \text{\hspace{0.17em}⁣}:\text{A}\to Q(\text{B})$. The relation between ${\text{Ext}}_u(\text{A},\text{B})$ and $\text{Ext}(\text{A},\text{B})$ is established. Using this relation, we show the half–exactness of ${\text{Ext}}_u(\cdot ,\text{B})$ and the (UCT) for ${\text{Ext}}_u(\text{A},\text{B})$. Furthermore, under certain conditions, we obtain the half–exactness and Bott periodicity of ${\text{Ext}}_u(\text{A},\cdot )$.