Outer automorphisms of separable C *-algebras (Q1098068)

Let A be a separable C *-algebra. Let Aut A be the group of *- automorphisms of A and let Inn A be the normal subgroup of automorphisms of A implemented by operators in the group U(A) of unitary operators in the multiplier algebra M(A) of A. Let \(Aut_ z A\) denote the normal subgroup of Aut A of automorphisms that pointwise fix the center Z of M(A). Let Aut A have the topology of pointwise uniform convergence. The space U(A) is a complete metric space with metric \[ d(u,v)=\sum 2^{- k}(\| (u-v)a_ k\| +\| a_ k(u-v)\|), \] where \(\{a_ k\}\) is a dense sequence of nonzero elements of A. The map ad: U(A)\(\to Inn A\subset Aut A\) is continuous and Inn A is a Borel subset of Aut A. If A does not have continuous trace, then Inn A is not closed in Aut A. In this case, the author shos that \(\overline{Inn A}/Inn A\) is not countably separated in the quotient Borel structure, and in particular, that \(Aut_ z A/Inn A\) is uncountable.