Automorphisms of C1-algebra bundles

Let B be a unital C¹-algebra and A = Γ(E) be the C¹-algebra of sections of a bundle over the separable compact space X with fibre B and structure group Inn B. If B is the quotient of an AW¹-algebra, then an exact sequence: 0 → Inn A → PInn A →^ηH²(X, G), where PInn A is the group of pointwise inner automorphisms of A and G=H^°(Z(B)^, $\text{Z}$) is obtained. The map η is onto whenever A = C(X, B) and B is the quotient of a purely infinite AW¹-algebra. An essential part of the analysis is the result that Ad: U(B) → Inn B is a fibre bundle if and only if the space of inner derivations of B is norm closed. These results extend and clarify previous joint work with I. Raeburn (Indiana Univ. Math. J.29 (1980), 799).