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Stream: theory: categorical probability

Topic: Noncommutative measurable spaces

Antonio Lorenzin (May 01 2025 at 15:06):

I'd like to advertise my recent paper: https://arxiv.org/abs/2504.13708, a joint work with @Tobias Fritz. Our goal has been to characterise what noncommutative measurable spaces are, by establishing a Gelfand-type duality for measurable spaces. We hope this might lead to interesting new directions in future research. A few important remarks about this work:

1. A standard misconception: aren’t noncommutative measurable spaces just von Neumann algebras?
   This is an important question, and understanding it also helps explain why the $C^*$-algebras we discuss may have been somewhat overlooked. Many people haven’t been interested in a categorical approach to measurable spaces, so the subtlety of the non-functorial connection between them and von Neumann algebras hasn’t gotten much attention. (See Pavlov’s work for a Gelfand-type duality for commutative von Neumann algebras: https://arxiv.org/abs/2005.05284.) This lack of interest makes sense to some extent—after all, we often end up considering a probability measure anyway. But Markov categories teach us that the measurable setting is crucial for describing determinism, conditioning, conditional independences, and almost sure equalities in a categorical way. We think von Neumann algebras remain very relevant, but a more general definition could offer deeper insights.

2. With the emerging notion of $C^*$-algebras, which we call $\sigma C^*$-algebras, we find that:

• $\sigma$-homomorphisms generalize PVMs, which in turn generalize measurable functions,
• $\sigma$-normal completely positive unital maps (all $C^*$-algebras are assumed unital) generalize POVMs, which in turn generalize Markov kernels.

3. Inspired by enveloping von Neumann algebras, we define Pedersen-Baire envelopes, which are associated to Baire $\sigma$-algebras in the commutative case. These can differ from Borel $\sigma$-algebras outside the separable case, and seem to be the “right” choice in several respects:

   • The Kolmogorov extension theorem can be extended to arbitrary infinite cardinalities,
   • Disintegration of measures still holds.
     This good behaviour had already been noticed by Jamneshan and Tao, who used the Baire setting to avoid separability restrictions.

Let me know what you think — I'm happy to talk more about it!

David Corfield (May 01 2025 at 15:16):

Does anything corresponding to a [[von Neumann algebra factor]] appear?

Benedikt Peterseim (May 01 2025 at 15:31):

Nice work! Does this connect to your paper on involutive Markov categories? (Sorry if it’s in the paper and I missed it.) Do you expect the semicartisian category mentioned in the final remark to fit into some abstract picture of “quantum Markov categories”?

Benedikt Peterseim (May 01 2025 at 15:32):

And will we finally know soon what the right way to look at “quantum logic” / “quantum programming semantics” is? :)

Antonio Lorenzin (May 01 2025 at 15:35):

@David Corfield The connection with von Neumann algebras and the correspondence to measure spaces are not discussed in the paper. However, there are promising results already in the literature: see Saito and Wright's book, as well as Pedersen's. These two books do not focus on $\sigma C^*$-algebras, but they employ them throughout. I honestly cannot tell you more at the moment, but for instance Saito and Wright do discuss factors, and they use a "generalized" notion of von Neumann algebras, i.e. one that does not require the separation by normal states.

Antonio Lorenzin (May 01 2025 at 15:43):

Benedikt Peterseim said:

Nice work! Does this connect to your paper on involutive Markov categories? (Sorry if it’s in the paper and I missed it.) Do you expect the semicartisian category mentioned in the final remark to fit into some abstract picture of “quantum Markov categories”?

The more I think about it, the more I think involutive Markov categories may not be the right framework :sweat_smile: (the topological closure for bounded maps was still ok, but $\sigma$-normality may be way more painful, and it does not seem to be worth it)

There are other possibilities to explore, such as multicategories, and douidal categories, but even the dilational perspective in semicartesian categories may have some interesting future prospects (check this out: https://arxiv.org/abs/2401.17447v1). But we could also consider explicitly the monad that I discussed in my previous paper (the one given by representability), which can be generalized to Pedersen-Baire envelopes, and use that to gain insight on things like "conditionals" and "determinism". In my personal opinion, one thing is important, and is actually underlying all the mentioned approaches: we should find a way to "encode" multiplication.

So concerning your second question, "soon" feels a bit of a stretch to me, but who knows :)