Lemma 44.4.3.label Let $(M, \tau)$ be a tracial von Neumann algebra in standard form, then
- (1)
For each $x, y \in \td M_{+}$, $\tau(x + y) = \tau(x) + \tau(y)$.
- (2)
For each $x \in \td M_{+}$ and $\mu \ge 0$, $\tau(\mu x) = \mu\tau(x)$.
- (3)
For each $x \in \td M_{+}$, $\tau(x^{*}x) = \tau(xx^{*})$.
Proof. (1): By Lemma 44.3.4, there exist non-decreasing sequences $\seq{p_n}, \seq{q_n}\subset \text{Proj}(M)$ such that:
- (i)
$p_{n}L^{2}(M, \tau) \subset D(x)$ and $q_{n}L^{2}(M, \tau) \subset D(y)$ for all $n \in \natp$
- (ii)
$\tau(p_{n}) \upto 1$ and $\tau(q_{n}) \upto 1$ as $n \to \infty$.
For each $n \in \natp$, let $r_{n} = p_{n} \wedge q_{n}$, then $r_{n}L^{2}(M, \tau) \subset D(x) \cap D(y) \subset D(x + y)$. Moreover, $\seq{r_n}\subset \text{Proj}(M)$ is non-decreasing, with $\tau(r_{n}) \upto 1$ as $n \to \infty$. As such, by Lemma 44.4.2,
(3): Let $x = u|x|$ be the polar decomposition of $x$, and $E: \cb_{[0, \infty)}\to B(L^{2}(M, \tau))$ be the resolution of the identity for $|x|^{2}$. By Proposition 44.3.2, $u \in U(M)$ and $E(B) \in M$ for all $B \in \cb_{[0, \infty)}$. For each $n \in \natp$, let $p_{n} = E([0, n])$, then by Lemma 44.4.2,
For each $n \in \natp$, let $q_{n} = up_{n}u^{*} \in M$ be the projection onto $up_{n}L^{2}(M, \tau)$. Then, $\tau(q_{n}) = \tau(up_{n}u^{*}) = \tau(p_{n} u^{*}u) = \tau(p_{n})$, so $\tau(q_{n}) \upto 1$ as $n \to \infty$. Therefore Lemma 44.4.2 implies that
$\square$
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