Proposition 44.2.4.label Let $(M, \tau)$ be a tracial von Neumann algebra, then:
- (1)
For each $x \in M$, $JL_{x}J = R_{x^*}$.
- (2)
$JL(M)J = R(M)$.
Proof. (1): For each $x, y \in M$,
\[JL_{x}J \wh y = JL_{x} \wh{y^*}= J \wh{xy^*}= \wh{yx^*}= R_{x^*}y\]
$\square$
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