Definition 44.2.1 (Standard Representation).label Let $(M, \tau)$ be a tracial von Neumann algebra, then the GNS representation
\[(\pi, L^{2}(M, \tau), \xi_{\tau})\]
is the standard representation of $(M, \tau)$. For each $x \in M$, identify $\wh x = x\xi_{\tau} \in L^{2}(M, \tau)$, so that $M = \wh M \subset L^{2}(M, \tau)$.
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