Lifting Automorphisms

Let A be a unital nuclear separable C*-algebra which satisfies the Universal Coefficient Theorem and E be a unital essential extension of the form: where is the C*-algebra of compact operators on l2. Suppose that Aut(E) is an automorphism on E. We show that if , the automorphism on A induced by , is in Aut0(A), the identity component of Aut(A), then is approximately inner if and only if an index ()=0. Consequently, in certain interesting cases, Aut0(E) if and only if idE] in KK(E,E) and is approximately inner if and only if idE] in KL(E,E). In particular, when K1(A) is torsion free, is approximately inner if and only if induces the identity map on K0(E).