# Trace-class, Hilbert Schmidt operators, $L^p(H)$: duality theorems

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.

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.