A $\mathcal{T}$ -algebras and extensions of A $\mathbb{T}$ -algebras

Abstract

Lin and Su classified A\( \mathcal{T} \)-algebras of real rank zero. This class includes all A\( \mathbb{T} \)-algebras of real rank zero as well as many C*-algebras which are not stably finite. An A\( \mathcal{T} \)-algebra often becomes an extension of an A\( \mathbb{T} \)-algebra by an AF-algebra. In this paper, we show that there is an essential extension of an A\( \mathbb{T} \)-algebra by an AF-algebra which is not an A\( \mathcal{T} \)-algebra. We describe a characterization of an extension E of an A\( \mathbb{T} \)-algebra by an AF-algebra if E is an A\( \mathcal{T} \)-algebra.