>>11265350

Families of sup-seminorms [math]N_C^\infty(x) = \sup_{y \in C}|y(x)|[/math] over compact sets [math]C\subset Y[/math] on locally convex spaces [math]X[/math] generates the Mackey topology [math]\tau(X,Y)[/math], so we can consider the "balls" obtained from t[math]N_C^\infty[/math].

By Mackey-Arens, any dual topology [math]\mathscr{T}[/math] is both coarser than the weak-operator topology [math]\sigma(X,Y)[/math] and finer than the Mackey topology [math]\tau(X,Y)[/math], which is in turn finer than the strong-operator topology [math]\rho(X,Y)[/math] (due to Bourbaki-Alaoglu/"balls are compact" theorem). We may wish, then, to investigate if we can still generate dual topologies with seminorms as we tune its coarse-ness between weak- and Mackey. Perhaps we need to change what the [math]C[/math]'s are in [math]Y[/math] or consider [math]N_C^p[/math] for [math]p<\infty[/math]. I'm not sure, but it certainly can be interesting.

>>11265377

Let's investigate this possibility in QM. Note that w- and s-closures of the algebra of observables coincide via von Neumann's bicommutant theorem. Since states are given by linear [math]*[/math]-irreps, weak and strong notions of convergence can only differ due to the topology on the rep-space, not that on the algebra.

Now when dealing with rigged reflexive Hilbert spaces with Frechet topology, such as tempered distributions, however, one can prove that closed bounded subsets in [math]\mathcal{H}'[/math] is compact, hence [math]\text{weak}\xrightarrow{\text{Frechet}}\text{Mackey} \rightarrow \text{strong}\rightarrow \text{weak}[/math]. So w- and s-convergence again coincide on Frechet spaces. In particular, since Stone-von Neumann tells you that there is essentially only one unitarily-equivalent irrep of the Weyl algebra, and Groenewold-van Hove quantization lands your rep-space in a maximal Poisson subalgebra of [math]C^\infty[/math] (which has Frechet topology), the strong-weak distinction basically never happens in QM.