Recovering the algebras of functions on a compact Hausdorff space, non-commutative analog Let $K$ be a compact-Hausdorff space and $\mu$ be a regular Borel measure. We have four important space to consider: the $C^*$ algebra of continuos functions $(C(K),||\cdot ||_{\infty})$, the abstract Von Neumann algebra of bounded measurable functions $(L^{\infty}(K),||\cdot||_{\infty})$, the space of integrable functions $(L^1(K),||\cdot||_1)$ (its predual) and the Hilbert space of square integrable functions $(L^2(K),||\cdot||_2)$.
Q1: $L^{\infty}$ depends only on the sets of measure zero $N$, can $L^1$ and $L^2$ (or $L^p$ in general) be defined only with $N$? Are there any restrictions on $N$ similar to $\mu$ to be regular?
We can recover $K$ from $A=C(K)$, and then also the spaces $W=L^{\infty}(K)$, $W_*=L^1(K)$ and $H=L^2(K)$.
Q2: Given a $C^*$ algebra $A$, can we associate a Hilbert space $H$ and a Von Neumann algebra $W$ so that in the case of $A=C(K)$, then $H=L^2(K)$, $W=L^{\infty}(K)$ and $W_*=L^1(K)$?
Q3: If so, what about associating to $A$, Banach spaces $B_p$ such that if $A=C(K)$ then $B_p=L^p(K)$?
Edit: Maybe in Q2 we have to consider a generalization of $N$, and for $B_p$ a generalization of a regular measure.
 A: A lot of serious research went into these questions, so I'll have to restrict myself mostly to giving some keywords. Feel free to ask for more details or references for concrete points.
Q1: Abstractly (that is, up to isometric isomorphism), the spaces $L^1(K)$ and $L^2(K)$ only depend on $N$. You can show this by writing down an explicit isomorphism using the Radon-Nikodym density of two measures with the same collection of measure zero sets. But more abstractly, $L^1(K)$ is the predual of $L^\infty(K)$ (as you wrote yourself) and $L^2(K)$ is the underlying Hilbert space of the standard form. Both are uniquely determined (up to isometric isomorphism) by the von Neumann algebra. The correct condition on $N$ is that the measure algebra be localizable.
But this is in some sense not the complete picture. The way $C(K)$ "sits" inside $L^1(K)$ and $L^2(K)$ does depend on the measure, that is, for different measures $\mu,\nu$ with the same collection of sets of measure zero, there is not necessarily a $\ast$-automorphism $\alpha\colon C(K)\to C(K)$ that is isometric with respect to the $L^1$- or $L^2$-norm.
Q2: As you noted, the von Neumann algebra depends on the choice of $N$. However, given a unital $C^\ast$-algebra $A$ and a state $\omega\colon A\to\mathbb{C}$, you can form the GNS Hilbert space $H_\omega$ and the weak closure of $A$ in the GNS representation. In the case when $A=C(K)$, every state is of the form $\omega=\int\cdot\,d\mu$ for some regular $\mu$ and you obtain $L^2(K)$ and $L^\infty(K)$ from this construction. I think it is hard to find a good analog of $N$ in the $C^\ast$-algebra picture because measurable sets don't fall in the realm of topology, but measure theory.
Q3: Again, von Neumann algebras seem more suitable here. Given a von Neumann algebra $M$ and a normal faithful state (or, more generally, norma semi-finite faithful weight) $\omega$ on $M$, one can form the noncommutative $L^p$-spaces $L^p(M,\omega)$. As it turns out, these spaces do not depend (up to isometric isomorphism) on the choice of $\omega$, although once again the interpolation scale does.
There are several different constructions of $L^p(M,\omega)$. The one that is probably easiest to describe uses interpolation. The space $L^2(M,\omega)$ is simply the GNS Hilbert space associated with $\omega$, which comes (if $\omega$ is a state) with a natural inclusion map $\Lambda_\omega\colon M\to L^2(M,\omega)$. Thus $(L^2(M,\omega),M)$ forms an interpolation couple and one defines $L^p(M,\omega)=(L^2(M,\omega),M)_{2/p}$ for $p\in (2,\infty)$. Moreover, $L^\infty(M,\omega)=M$, $L^1(M,\omega)=M_\ast$ and $L^p(M,\omega)=L^q(M,\omega)^\ast$ for $p\in (1,2)$ and $1/p+1/q=1$.
