A characterization of thin operators in a von Neumann algebra

Abstract:

Let $\mathcal {A}$ be a von Neumann algebra, $\mathcal {J}$ a uniformly closed, weakly dense, two-sided ideal in $\mathcal {A},\mathcal {L}$ the center of $\mathcal {A}$, and $\mathcal {P}$ the lattice of projections in $\mathcal {J}$. An operator $A \in \mathcal {A}$ is thin relative to $\mathcal {J}$ if $A = Z + K$, for some $Z \in \mathcal {L},K \in \mathcal {J}$. The thin operators relative to $\mathcal {J}$ are characterized as those $A \in \mathcal {A}$ satisfying ${\lim _{P \in \mathcal {P}}}||AP - PA|| = 0$. It is also shown that \[ \lim \sup \limits _{P \in \mathcal {P}} ||PAP - AP|| = \lim \sup \limits _{P \in \mathcal {P}} ||PAP - PA||.\]