---
tag: CA38T5ZH14
kind: concept
name: Ultraweak topology
msc: [46L10, 47L05]
sources:
  - url: https://en.wikipedia.org/wiki/Ultraweak_topology
same_as:
  wikidata: Q7880722
  wikipedia: Ultraweak topology
---

## Summary

The ultraweak topology, also called the $\sigma$-weak topology, is a topology on the bounded operators $B(H)$ on a Hilbert space $H$.
It is the weak-$*$ topology that comes from viewing $B(H)$ as the dual of the space of [[trace class operators|Z2CP530WDE]].
It is finer than the [[weak operator topology|4XM1VN1C1N]] but agrees with it on bounded sets.
The linear functionals that are continuous for it are exactly the normal functionals, which makes it the intrinsic topology of a [[von Neumann algebra|PSFWQVZ0FS]].

## Explanations

### L1 human

The weak operator topology tests an operator $T$ against finitely many pairs of vectors at a time.
The ultraweak topology allows infinitely many pairs at once, as long as the total size of the vectors is finite.
This is the same as testing $T$ against density matrices $\rho$ through the number $\operatorname{tr}(T \rho)$, the expected value of $T$ in the state $\rho$ in quantum mechanics.
On the unit ball nothing changes, but globally the topology has more open sets.
Its advantage is that it depends only on the algebra and not on the Hilbert space on which it acts.