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

TL;DR

This paper investigates the structure of the group of unitary equivalence classes of unital extensions of certain nuclear C*-algebras, establishing relations with existing Ext groups and proving properties like half-exactness and Bott periodicity.

Contribution

It introduces the group xt of unital extensions, relates it to xt, and proves new properties such as half-exactness and Bott periodicity for these groups.

Findings

Established the relation between xt(\u00a0A,) and xt(,)

Proved half-exactness of xt(,)

Proved Bott periodicity under certain conditions

Abstract

Let $\A$ be a unital separable nuclear $C^*$--algebra which belongs to the bootstrap category $\N$ and $\B$ be a separable stable $C^*$--algebra. In this paper, we consider the group $\Ext_u(\A,\B)$ consisting of the unitary equivalence classes of unital extensions $\tau\colon\A\rightarrow Q(\B)$. 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 $\Ext_u(\A,\cdot)$.