Classification of Extensions of A𝕋-Algebras

Abstract

We classify certain extensions of A𝕋-algebras using the six-term exact sequence in K-theory together with the Elliott invariants of the ideal and quotient. We also give certain necessary and sufficient conditions for such extension algebras being A𝕋-algebras.

Classification of unital extensions ofAT-algebras

“…M. Rordam's result plays a crucial role in this field since he firstly classified extensions up to stable isomorphism with six-term exact sequences and KK -groups. Subsequently, Eilers et al [10] classified stable full extensions of certain classifiable C * -algebras and we [11] gave a classification theorem for non-unital full extensions of AT-algebras. Most of these results have commonness: they only dealt with non-unital extensions and classification theorems were presented in the sense of stable isomorphism.…”

“…Lin and Rordam [7] have solved the case of AT-algebras with real rank zero. We [11] have proved that this question is equivalent to when extensions are quasidiagonal in the case of non-unital full extensions. In this note, using stabilization extensions and results for the case of nonunital extensions we prove that a similar result holds for unital full extensions of AT-algebras (see Theorem 4.2).…”

Unital extensions of AF-algebras by purely infinite simple algebras

“…Among these algebras, extension algebras are an important class. The existing results for classification of such algebras mainly focus on classification of non-unital extensions up to stable isomorphism, for example, [3], [17], [21].…”

“…As the succeeding work of [22], [21], [20], [23], [24], the purpose of this note is to classify unital essential extensions of AF -algebras by stable purely infinite simple algebras. Using the classification theory of C * -algebras and the universal coefficient theorem for unital extensions obtained by the second author ( [22], [23]), we give a complete characterization of isomorphisms between unital extensions of AFalgebras by stable Cuntz algebras.…”

Classification of extensions of torus algebra II

“…So one has to study isomorphism equivalence of extensions. There are many classification results of such extension algebras (see [2,9,14,17,[20][21][22], etc. ).…”