Definition 45.1.1 (Trace).label Let $H$ be a complex Hilbert space, $M \subset B(H)$ be a von Neumann algebra, and $\tau: M_{+} \to [0, \infty]$, then $\tau$ is a trace on $M$ if:

1. (1)

   For each $x, y \in M_{+}$ and $\lambda \ge 0$, $\tau(\lambda x + y) = \lambda \tau(x) + \tau(y)$.

2. (2)

   For each $x \in M_{+}$ and $u \in U(M)$, $\tau(uxu^{*}) = \tau(x)$.