Locally inner automorphisms of operator algebras

Details Locally inner automorphisms of operator algebras Locally inner automorphisms of operator algebras

2006-09-26

arxiv.org/abs/math/0609735v2

Mathematics

Operator Algebras

17 pages; some substantive changes to the last section ("problems and comments")

Scientific paper

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-Stormer. 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 descends to a well-defined map on the automorphism orbits of a unital C*-algebra if and only if all its automorphisms are locally inner.

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