Lemma 44.4.8.label Let $(M, \tau)$ be a tracial von Neumann algebra and $p \in [0, \infty]$, then:
- (1)
Positive simple elements of $M$ are dense in $L^{p}(M, \tau)_{+}$.
- (2)
Simple self-adjoint elements of $M$ are dense in $L^{p}(M, \tau)_{sa}$.
- (3)
$M$ is dense in $L^{p}(M, \tau)$.
Proof. (1): For each $z \in \td M_{+}$, let $E(z, \cdot): \cb_{[0, \infty)}\to M$ be the resolution of the identity for $z$.
Let $x \in L^{p}(M, \tau)_{+}$. For every $n \in \natp$, let $p_{n} = E(x, [0, n])$, then $E(xp_{n}, d\lambda) = E(x, \one_{[0, n]}d\lambda)$ and $E(x(1 - p_{n}), d\lambda) = E(x, \one_{(n, \infty)}d\lambda)$. By the Dominated Convergence Theorem,
As $\seq{xp_n}\subset M_{+}$, $M_{+}$ is dense in $L^{p}(M, \tau)_{+}$. By Theorem 43.7.2, positive simple elements of $M$ are dense in $L^{p}(M, \tau)_{+}$.
(2): By (1) applied to $0 \vee x$ and $-(0 \wedge x)$.$\square$
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