Definition 12.3.19 (Factors).label We have finite dimensional tracial vNAs, $M_{n}(\com)$ equipped with $\frac{1}{n}Tr=:\tau$. A $II_{1}$ factor is an infinite dimensional tracial vNA whose centre is reduced to the scalar operator.

A factor is of type $I$ if it is isomoprhic to $\cali{B}(\cali{H})$

Let $(X,\mu)$ be a probability space, $E$ countable Borel equivalence relation on $X$ then $E$ has countable classes and $E\suf X^{2}$ is Borel. We define the measure $\nu$ on $E$ as

then $L^{2}(E,\nu$) is a Hilbert space and $\nu$ is $\s$-finite measure as a consequence of Luzin. We define $\cali{M}_{b}(E)$ to be bounded finitely supported Borel functions $f:E\to\com$ where $\exists C>0,\fall x,y$ we have $\abs{z\in X:F(z,y)\neq 0},\abs{z\in X:F(x,z)\neq 0}\leq C$. Denote $B_{b}(X)$ the bounded Borel functions on $X$ then $f\mto f(x)\indi{\la}$ is an injection $B_{b}(X)\inj \cali{M}_{b}(E)$. $E$ is probabiliity measure preserving (pmp) if for any Borel automorphism $f:X\to X$ such that $xEf(x),f_{*}\mu=\mu$.