Abstract:
We consider a von Neumann algebra $M$ acting on a
Hilbert space $H$. For a positive operator $X$ in $M$ we define the
operator ``intervals'' $I_X=\{Y=Y^*\in M: \; -X \leq Y \leq X \}$ and
$L_X=\{Y \in M: \; |Y| \leq X \}$, where $|Y|=\sqrt{Y^*Y}$.
The properties of this operator ``intervals'' are investigated.
We prove that a von Neumann algebra $M$ is Abelian if and only if
$L_X$ is convex for all $X$ in $M$. We then show for $M=B(H)$, the algebra of all linear bounded
operators on $H$, that the operator ``interval'' $I_X$ is compact if and only if an operator $X$ is compact.
