Locally Inner Automorphisms of Operator Algebras

In this paper an automorphism of a unital C-algebra is said to be locally inner if on any element it agrees with some inner automorphism. We make a fairly complete study of local innerness in von Neumann algebras, incorporating comparison with the pointwise innerness of Haagerup-Størmer. On some von Neumann algebras, including all with separable predual, a locally inner automorphism must be inner. But a transfinitely recursive construction demonstrates that this is not true in general. As an application, we show that the diagonal sum (x, y) 7→ ( x 0 0 y ) descends to a welldefined map on the automorphism orbits of a unital C-algebra if and only if all its automorphisms are locally inner. “ ‘The inner truth is hidden — luckily, luckily. But I felt it all the same. . . .’ ” (Joseph Conrad, Heart of Darkness)