The unital Ext-groups and classification of $C^\ast$-algebras
James Gabe, Efren Ruiz

TL;DR
This paper develops a $K$-theoretic classification framework for unital and non-unital extensions of $C^*$-algebras using unital $ ext{Ext}$-groups, providing a complete classification for certain classes of algebras.
Contribution
It introduces the concept of unital $ ext{Ext}$-groups and applies them to classify extensions of $C^*$-algebras, including a full classification of extensions of UCT Kirchberg algebras by stable AF algebras.
Findings
Complete $K$-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.
Identification of invertible elements in semigroups of unital extensions as unital $ ext{Ext}$-groups.
Application of unital $ ext{Ext}$-groups to classify both unital and non-unital extensions.
Abstract
The semigroups of unital extensions of separable -algebras come in two flavours: a strong and a weak version. By the unital -groups, we mean the groups of invertible elements in these semigroups. We use the unital -groups to obtain -theoretic classification of both unital and non-unital extensions of -algebras, and in particular we obtain a complete -theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.
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Taxonomy
TopicsAdvanced Operator Algebra Research · Advanced Topics in Algebra · Algebraic structures and combinatorial models
