Normalizers of Operator Algebras and Reflexivity
Abstract
The set of normalizers between von Neumann (or, more generally, reflexive) algebras A and B, (that is, the set of all operators x such that xAx* is a subset of B and x*Bx is a subset of A) possesses `local linear structure': it is a union of reflexive linear spaces. These spaces belong to the interesting class of normalizing linear spaces, namely, those linear spaces U for which UU*U is a subset of U. Such a space is reflexive whenever it is ultraweakly closed, and then it is of the form U={x:xp=h(p)x, for all p in P}, where P is a set of projections and h a certain map defined on P. A normalizing space consists of normalizers between appropriate von Neumann algebras A and B. Necessary and sufficient conditions are found for a normalizing space to consist of normalizers between two reflexive algebras. Normalizing spaces which are bimodules over maximal abelian selfadjoint algebras consist of operators `supported' on sets of the form [f=g] where f and g are appropriate Borel functions. They also satisfy spectral synthesis in the sense of Arveson.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 May 2000
 arXiv:
 arXiv:math/0005178
 Bibcode:
 2000math......5178K
 Keywords:

 Mathematics  Operator Algebras;
 47L05 (Primary) 47L35;
 46L10 (Secondary)
 EPrint:
 20 pages