Throughout, let $(\mathcal{M},\tau)$ be a von Neumann algebra $\mathcal{M}$, acting on a Hilbert space $H$, with normal semifinite faithful trace $\tau$.
In the survey article by Pisier and Xu, the non-commutative $L^p$ space $1\leq p<\infty$ is defined as follows. Let $S_+$ be the set of all positive elements $x\in \mathcal{M}$ whose support projection $\text{supp}(x)$ has finite trace. If $\mathcal{S}$ is the linear span of $S_+$, then the non-commutative $L^p$ space $L^p(\mathcal{M},\tau)$ is just the completion of $\mathcal{S}$ with respect to the norm $$|| x ||_p:=[\tau(|x|^{p})]^\frac{1}{p}.$$
On the other hand, for instance in Nelson, an approach analogous to the construction of the classical $L^p$ spaces in measure theory is used. Briefly, we introduce the concept of a $\tau$-measurable operator. These are, by definition, closed densely defined operators $x$ on $H$ affiliated with $\mathcal{M}$ and satisfy the condition $$\tau(E_\lambda(|x|)<\infty,$$ where $E_\lambda(|x|)$ is the spectral resolution of $|x|$ corresponding to the indicator function of $(\lambda,\infty)$. By placing a specific topology on the set of all $\tau$-measurable operators, we obtain a metrizable topological $*$-algebra, denoted $L^0(\mathcal{M},\tau)$. After extending the trace $\tau$ to $L^0(\mathcal{M},\tau)$, we then define for $0<p<\infty$ $$L^p(\mathcal{M},\tau):=\{x\in L^0(\mathcal{M},\tau)|\ \tau(|x|^p)<\infty\}.$$
I have two questions:
- Why are these two approaches in defining the non-commutative $L^p$ spaces ($1\leq p<\infty$) equivalent? Are there any references to proof(s) of this fact?
- The space $\mathcal{S}$ above turns out to be weak$^*$ dense in $\mathcal{M}$. Will the completion of $(\mathcal{S},|| \cdot||_p$) still yield the non-commutative $L^p$ space if we assume that $\mathcal{S}$ is dense in a different operator topology?
