Lifting Unitary Elements with Application to Approximately Inner Automorphisms

Abstract

Let A be a separable unital nuclear simple C*-algebra with torsion K ₀(A), free K ₁(A) and with the UCT. Let τ : A→M(${\fancyscript K}$)/${\fancyscript K}$ be a unital homomorphism. We prove that every unitary element in the commutant of τ (A) is an exponent, thus it is liftable. We also prove that each automorphism α on E with $\ifmmode\expandafter\bar\else\expandafter\=\fi{\alpha } \in {\text{Aut}}_{0} {\left( A \right)}$ is approximately inner, where E is a unital essential extension of A by ${\fancyscript K}$ and $\ifmmode\expandafter\bar\else\expandafter\=\fi{\alpha }$ is the automorphism on A induced by α.