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.

Let A be a unital separable nuclear C * -algebra and let B be a stable C * -algebra. Using K-theory and KK-theory we establish universal coefficient theorems for the stable Ext-groups of unital extensions of A by B when A and B have certain properties, which generalize a result of L. Brown and M. Dadarlat for the strong Ext-groups. The class of extensions being studied are also enlarged.

Characterizations of hereditary subalgebras generated by subsets of a C*-algebra are given through open projections. Using these results, we give some equivalent conditions of comparison of positive elements.

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We use extension theory and algebraic methods to give a complete characterization of extensions of torus algebra by stable Cuntz algebras, and prove certain classification theorems of these extension algebras.