Let $H$ be a Hilbert space, separable if necessary, and let $tr$ be the usual trace on $L^1(H)$. It is classical theory that $K(H)^*=L^1(H)$, and $L^1(H)^*=B(H)$, via the canonical application $\langle T,S\rangle=tr(TS)$ for suitable $T$ and $S$.
It is often mentioned that these duality results are analogous to the well-known $c_0^*=\ell_1$ and $\ell_1^*=\ell_\infty$. I think I more or less understand how this analogy works: The operator norm in $B(H)$ is the uniform norm when we restrict the maps to the unit ball, so $B(H)$ consists of the maps with finite "uniform norm", $K(H)$ consists of the elements which can be approximated by finite rank ones, and these are clear analogues of $\ell_\infty$ and $c_0$, respectively. I see more or less the analogy between $L^1(H)$ and $\ell_1$: identify the $i$-th element of a fixed basis for $H$ with the $i$-coordinate function in $\ell^1$, but I don't really see, precisely, how $\langle T\xi_i,\xi_i\rangle$ relates with $x_i$, for $T\in L^1(H)$ and $x=(x_i)\in\ell_1$.
My question is what is the precise relation between these two duality results. More precisely,
Question 1: Can we obtain the duality results $c_0^*=\ell_1$ and $\ell_1^*=\ell_\infty$ from $K(H)^*=L^1(H)$ and $L^1(H)^*=B(H)$? What if we consider a measure/probability space $(X,\mu)$ instead of $\mathbb{N}$ with counting measure (although $c_0$ has no analogue in this case).
More generaly, we might define $L^p(H)$ to consist of those $T\in B(H)$ for which the "norm" $\Vert T\Vert_p=(\Vert |T|^p\Vert_1)^{1/p}$ is finite. I'm not sure if this is indeed a complete norm, but if we assume so, it is natural to ask:
Question 2: Let $p$ and $q$ be (finite) Hölder conjugates. Are $L^p(H)$ and $L^q(H)$ duals of each other (with the usual application $\langle T,S\rangle=tr(TS)$), and does this indeed generalize the duality of $\ell^p$ and $\ell^q$? Moreover, is the map $p\mapsto\Vert T\Vert_p$ continuous in some sense and does $\Vert T\Vert_p$ converges to $\Vert T\Vert$ as $p\to \infty$ for $T\in \cap_p L^p(H)$?
I'm interested in these questions to understand better how tracial von Neumann algebras/$II_1$ factors are related to probability spaces, and the importance of the Hilbert-Schmidt norm on such algebras.
Most of your question is answered by Theorem 3.2 in Simon's Trace Ideals and Their Applications. It is not an easy read because of the awkward notation, but it is worth it.
Yes, $L^q(H)$ is the dual of $L^p(H)$, basically because Hölder holds, $|\text{Tr}(TS)|\leq\|T\|_p\|S\|_q$. I cannot say that the dualities in the sequence spaces follow from the dualities in $B(H)$, but you'll find (the first half of) Chapter 2 in Simon's book interesting, where Lidskii's results are developed. I have no idea about going from $\mathbb N$ to $X$.
The convergence of the $p$-norms follows in a rather obvious way from $\|T\|=\mu_1(T)$ and $$ \|T\|_p=\left(\sum_{k=1}^\infty\mu_k(T)^p\right)^{1/p} $$ (where $\mu_k(T)$ are the singular values of $T$) with the same proof as in the discrete measure case.