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/Part 8: Von Neumann Algebras/Chapter 44: Tracial von Neumann Algebras/Section 44.4: Noncommutative $L^{p}$ Spaces

Theorem 44.4.5.label Let $(M, \tau)$ be a tracial von Neumann algebra in standard form, $L^{2}(M, \tau)$ denote the Hilbert space associated with its standard form, and $\mathcal{L}^{2}(M, \tau)$ be the space of square integrable operators, then the mapping

\[L^{2}(M, \tau) \to \mathcal{L}^{2}(M, \tau) \quad \xi \mapsto L_{\xi}\]

is an isometric isomorphism.

Proof, [Theorem 7.3.2, AP]. (Well-Defined): Let $\xi \in L^{2}(M, \tau)$ and $u \in U(M') = U(R(M))$. By Theorem 44.2.8, there exists $v \in U(M)$ such that $u = R_{v}$. In which case, for each $x \in M$,

\[R_{v} L_{\xi} R_{v}^{*} \wh x = R_{u} L_{\xi} \wh{xu^*}= R_{u} R_{xu^*}\xi = R_{xu^*u}\xi = R_{x}\xi = L_{\xi} \wh x\]

so $L_{\xi}^{0}$ is affiliated with $M$. By Lemma 44.3.7, $L_{\xi}$ is also affiliated with $M$.

(Isometric): Let $\xi \in L^{2}(M, \tau)$ and $E: \cb_{[0, \infty)}\to M$ be the resolution of the identity for $|L_{\xi}|$. By Definition 43.8.12, $D(|L_{\xi}|) = D(L_{\xi})$. In particular, $\wh 1_{M} \in D(L_{\xi})$, so

\begin{align*}\norm{L_\xi}_{\mathcal{L}^2(M, \tau)}&= \int_{[0, \infty)}\lambda^{2} E_{\wh 1_M, \wh 1_M}(d\lambda) = \normn{|L_\xi|\wh 1_M}_{L^2(M, \tau)}^{2} \\&= \normn{L_\xi \wh 1_M}_{L^2(M, \tau)}^{2} = \norm{\xi}_{L^2(M, \tau)}^{2}\end{align*}

(Surjective): Let $T \in \mathcal{L}^{2}(M, \tau)$ and $\xi = T \wh 1_{M}$, then for each $x \in M$,

\[L_{\xi} \wh x = R_{x} T \wh 1 = TR_{x} \wh 1_{M} = T \wh x\]

so $L_{\xi} \subset T$. By Proposition 44.3.3, $L_{\xi} = T$.$\square$

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