Definition 44.4.1 (Extension of Trace).label Let $(M, \tau)$ be a tracial von Neumann algebra in standard form. For each positive element $x \in \td M_{+}$, let $E(x, \cdot): \cb_{[0, \infty)}\to M$ be its resolution of the identity, then $\tau$ extends to $\td M_{+}$ as follows:
\[\tau: \td M_{+}\to [0, \infty] \quad \int_{[0, \infty)}\lambda E(x, d\lambda)_{\wh 1_M, \wh 1_M}\]
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