Extensions for AF 𝐶* algebras and dimension groups

Source abstract

Let A A , C C be approximately finite dimensional ( AF) C ∗ ({\text {AF)}}\,{C^{\ast }} algebras, with A A nonunital and C C unital; suppose that either (i) A A is the algebra of compact operators, or (ii) both A A , C C are simple. The classification of extensions of A A by C C is studied here, by means of Elliott’s dimension groups. In case (i), the weak Ext group of C C is shown to be Ext Z ( K 0 ( C ) , Z ) {\operatorname {Ext} _{\mathbf {Z}}}({K_0}(C),\,{\mathbf {Z}}) , and the strong Ext group is an extension of a cyclic group by the weak Ext group; conditions under which either Ext group is trivial are determined. In case (ii), there is an unnatural and complicated group structure on the classes of extensions when A A has only finitely many pure finite traces (and somewhat more generally).