Some results on $A\mathcal{T}$-algebras are given. We study the problem when ideals, quotients and hereditary subalgebras of $A\mathcal{T}$-algebras are $A\mathcal{T}$-algebras or $A\mathcal{T}$-algebras, and give a necessary and sufficient condition of a hereditary subalgebra of an $A\mathcal{T}$-algebra being an $A\mathcal{T}$-algebra.

In this paper we consider the question when a homomorphism between two extension algebras preserves the essential ideal in the corresponding extension. Some conditions of two essential extensions being isomorphic are given. We also describe the relationship between the induced extensions and the Kasparov products and give the completely positive liftings of the induced extensions.

In this paper, we consider equivalence relations of $C^∗$-algebra extensions and describe the relationship between the isomorphism equivalence and the unitary equivalence. We also show that a certain group homomorphism is the obstruction for these equivalence relations to be the same.

We discuss properties of quotient semigroup of abelian semigroup from the viewpoint of $C^∗$-algebra and apply them to a survey of extension semigroups. Certain interrelations among some equivalence relations of extensions are also considered.