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The Quarterly Journal of Mathematics Advance Access published online on January 9, 2009
The Quarterly Journal of Mathematics, doi:10.1093/qmath/han039
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© 2009. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

LOCALLY INNER AUTOMORPHISMS OF OPERATOR ALGEBRAS

David Sherman{dagger}

Department of Mathematics, University of Virginia, PO Box 400137, Charlottesville, VA 22904, USA

{dagger} Corresponding author. E-mail: dsherman{at}virginia.edu

Received 7 March 2008; revised 8 December 2008
  Abstract

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 Formula 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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