7. Von-Neumann alghebra

7.1. Introduction and Definitions

We begin with a quick review definitions of concepts needed here. Let $H$ be a Hilbert space. Recall that a von-Neumann algebra $M$ acting on $H$ is a unital selfadjoint subalbegra of the algebra $P\left[B\right]\left(H\right)$ of all bounded linear operators on $H$. Alternatively, a von-Neumann algebra can be defined as a $C^{\ast }-algebra$ with a predual. By the predual of a Banach algebra $B$ we mean a Banach space $B_{\ast }$ whose dual is $B$. For more extensive definitions of concepts and results, [see ??????], etc. or see the excellent wikipedia-article ’Von Neumann algebra’.

In the following $M$ will denote a von Neumann algebra unless otherwise specified.