A characterization of thin operators in a von Neumann algebra

Catherine L. Olsen
1973 Proceedings of the American Mathematical Society  
Let si be a von Neumann algebra, J a uniformly closed, weakly dense, two-sided ideal in si, S the center of si', and 9 the lattice of projections in J. An operator A e si is thin relative to J if A =Z+K, for some Ze2C,KeJ. The thin operators relative to J are characterized as those A es/ satisfying \\mpv?\\AP-PA\\=0. It is also shown that lim sup \\PAP -AP\\ = lim sup \\PAP -PA\\. Pe& PeSP 1. Let si be a von Neumann algebra, «/ a uniformly closed, weakly dense, two-sided ideal in si, 2£ the
more » ... er of si, and SP the lattice of projections in J. An operator A e si is í«¡« relative to J if ^=Z+A^, for some ZeJ and KeJ. The set ^ is directed under the usual ordering, and for a fixed A e j/, the map P-y\\AP-PA\\ is a net on ^. In Theorem 1 we characterize the thin operators relative to J^ as those A e sé satisfying lim \\AP -PA\\ =0. This discussion is a sequel to a previous paper [7] in which a version of this characterization was demonstrated for the case of si a factor. The proof to be presented here requires results of the previous paper. It was proved in [7] that for si a factor, A e si is thin relative to J" if and only if lim \\PAP -AP\\ = 0. Pea» This was done by R. G. Douglas and Carl Pearcy for si=¿M(J4f), the algebra of all bounded operators on a separable Hubert space 3*i?, and J the ideal of compact operators [3] . It is an immediate corollary that for
doi:10.1090/s0002-9939-1973-0341121-5 fatcat:kzysobhqt5erzpmto5wpv7rr6y