Project idea: Trace class operators

Stream: LFTCM 2024

[Frédéric Dupuis]

One possible project is to define trace-class operators. I have created a repository here to get started.

[Frédéric Marbach]

I'm interested.

[Kevin Buzzard]

I trust that you'll be working in the right generality, so we can use the concept for p-adic Banach spaces! Serre wrote a paper on this stuff in around 1968 (it has applications to the Weil conjectures)

[Jireh Loreaux]

Kevin, can you be more specific? Do you mean defining nuclear operators?

[Jireh Loreaux]

@Kevin Buzzard can you expand?

[Kevin Buzzard]

In the nonarchimedean case Hilbert-Schmidt and trace class operators coincide, they are both the limits of finite rank operators. My point is just that they're an important concept in the p-adic case.

[Jireh Loreaux]

Okay, limits in what norm (or topology)?

[Antoine Chambert-Loir]

Kevin is point out to this paper of Serre. http://www.numdam.org/item/PMIHES_1962__12__69_0/

[Antoine Chambert-Loir]

(Over a p-adic field, it's the same for a series to be summable or to converge to 0, hence the delicate hierarchy of operators that exists in the real world collapses.)

[Anatole Dedecker]

I don’t know much about this, but I think it’s highly non obvious wether it’s even possible to factor any code here

[Kevin Buzzard]

ooh was it really 62? I generalised everything to Banach modules over Banach rings in the nonarch case at the beginning of this paper here https://www.ma.imperial.ac.uk/~buzzard/maths/research/papers/eigenvarieties.pdf (Coleman had done this first but his exposition contained some slips).

[Antoine Chambert-Loir]

It is not clear to me whether real and p-adic functional analysis should be made at the same time. And after all, a good theory of a spectrum of a compact operator requires Berkovich spaces.

[Kevin Buzzard]

Yes you might well be right: press on with the real/complex story if you need it, and the p-adic people will catch up later. In the p-adic world everything is far less delicate, as I'm sure you can imagine in a situation where an infinite sum converges iff its terms tend to zero.

[Jireh Loreaux]

Yes, in the bounded operators on a Hilbert space, there is an extremely rich ideal structure given by the Calkin correspondence. Basically, the two-sided ideals are in bijective correspondence (even an isomorphism of ordered semirings) with the dilation invariant, hereditary subcones of the nonnegative sequences converging to zero. I think there will be very little overlap, but I'm happy to chat about it at some point.

I happen to know a lot about this topic because my advisor was Gary Weiss, who coauthored this paper, which is basically the bible for ideal theory in bounded operators on a Hilbert space.

[Jireh Loreaux]

@Frédéric Dupuis @Frédéric Marbach : I just pushed to the repo the continuous functional calculus over ℝ≥0 for H →L[ℂ] H.