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Stream: learning: questions

Topic: L^p as adjoint

Leopold Schlicht (May 04 2021 at 19:09):

Can one describe $L^p$ as (left or right) adjoint functor, similar to how $\ell^1$ (from $\mathbf{Set}$ to the category of Banach spaces and operators with norm $\leq 1$) is left adjoint to $\hom(K,-)$ (the unit ball functor) and $\ell^\infty$ is right adjoint to $\hom(-,K)$?

Fawzi Hreiki (May 04 2021 at 19:31):

There is this by Tom Leinster

Leopold Schlicht (May 05 2021 at 12:18):

That's certainly related! But it characterizes the $L^p$ spaces as initial objects, rather than describing $L^p$ as an adjoint functor.

John van de Wetering (May 05 2021 at 13:54):

I believe I saw a paper about this exact thing a while back. I'm gonna see if I can dig it up

John van de Wetering (May 05 2021 at 13:56):

Although perhaps I'm mistaken and it was the Tom Leinster paper

John van de Wetering (May 05 2021 at 14:00):

Okay I can't find it (it doesn't help that the terms "adjoint" and "duality" are extremely overloaded). I believe it had something to do with Hölders inequality (relating L^p spaces and L^q spaces with 1/p+1/q = 1), but maybe I'm just hallucinating all of this.

John Baez (May 05 2021 at 14:15):

Of course initial objects are adjoint functors: the initial object in a category C is left adjoint to the functor from C to the terminal category. But maybe that's not the kind of adjoint functor you wanted!

Leopold Schlicht (May 05 2021 at 16:26):

@John van de Wetering Thanks nevertheless for the effort! :-)
@John Baez Yes, I'm talking about $L^p$ (as a functor from some category of measure spaces to some category of Banach spaces) being an adjoint functor, not each $L^p(X, \mathcal A, \mu)$ separately. :P

John van de Wetering (May 05 2021 at 22:43):

Not exactly what you are thinking of, but perhaps still of interest to you is this paper: http://arxiv.org/abs/2005.05284
He finds a Gelfand-type duality for commutative von Neumann algebras, relating them to a type of measure spaces (see Theorem 1.1 on page 2).

Leopold Schlicht (May 07 2021 at 19:56):

Thanks!

Dmitri Pavlov (May 08 2021 at 03:05):

Yes, you can. The point is to consider all L^p at once. Replacing p by 1/p, the spaces L^p become a complex-graded commutative *-algebra, in fact, an extended von Neumann algebra. This extended von Neumann algebra is freely generated by L^0 (= bounded measurable functions) in degree 0 and L^1 (= finite measures) in degree 1, where L^0 is considered as a von Neumann algebra and L^1 is a module over L^0.

Dmitri Pavlov (May 08 2021 at 03:07):

A concrete expression of the above fact: any element of L^p can be presented in the form f μ^p, where f∈L^0 (or rather, the extended L^0, allowing for unbounded functions) and μ∈L^1 is positive. The freeness condition says this presentation is unique up to the obvious relation: f (gμ)^p = fg^p μ^p, for any g∈L^0.

John Baez (May 08 2021 at 15:53):

Cool! So your $L^0$ is the usual $L^\infty$, right?

Dmitri Pavlov (May 08 2021 at 17:50):

Yes, exactly. The best way to think of positive elements of L^p is that they are precisely pth powers of positive elements of L^1, i.e., (positive) finite measures. So if μ is a (positive) finite measure, then μ^p ∈ L^p. Any element of L^p (if Re p > 0) can be uniquely presented in the form f μ^p, where μ is a positive finite measure, f is a U(1)-valued function (if Im p = 0; but not otherwise), and f and μ have the same support. For p=1 this is just the Radon–Nikodym theorem for complex-valued finite measures. If Re p = 0, then any element of L^p has a nonunique presentation in the form f μ^p, where μ is a positive measure (no longer necessarily finite), f≥0, and f and μ have the same support.

Dmitri Pavlov (May 08 2021 at 17:53):

All of this extends to noncommutative von Neumann algebras, where L^p for imaginary p encodes precisely the Tomita-Takesaki modular theory. Specifically, if μ is a semifinite normal weight on a von Neumann algebra M (commutative or not) and x∈M, then μ^{it} x μ^{-it} equals σ_μ^t(x), the Tomita–Takesaki modular automorphism of μ applied to x. Here μ^{it} ∈ L^{it}(M), the imaginary L^p-space of M, so μ^{-it} ∈ L^{-it}(M) and μ^{it} x μ^{-it} ∈ L^0(M) since x∈L^0(M).

Leopold Schlicht (May 08 2021 at 18:16):

Thanks!
But I have to admit that I don't understand what the adjoint functor of $L^p$ is after all.
I know nothing about *-algebras, von Neumann algebras, and so on. :P

Dmitri Pavlov (May 08 2021 at 18:30):

A \-algebra is a complex algebra with a complex-antilinear operation \: A→A such that (ab)\=b\a\* and 1\*=1. (Backslash does not escape asterisks properly on Zulip.)
For algebras of complex-valued functions, * is simply the conjugation operation. A commutative von Neumann algebra is precisely the algebra of (equivalence classes of) bounded measurable functions on a localizable enhanced measurable space (https://arxiv.org/abs/2005.05284).

Dmitri Pavlov (May 08 2021 at 18:40):

The adjunction is formulated as follows. The category C has as its objects pairs (M,X), where M is a commutative von Neumann algebra (equivalently, a compact strictly localizable enhanced measurable space S) and X is a *-module over M (equivalently, a measurable bundle over S). Morphisms (M,X)→(M',X') are morphisms M→M' of von Neumann algebras together with a morphism X→X' of *-modules that preserves supports. The category D is the category of C-graded extended von Neumann algebras. The right adjoint D→C to L sends a C-graded extended von Neumann algebra to its degree 0 and 1 components, where the degree 0 component is a commutative von Neumann algebra M and the degree 1 component is a *-module X over this commutative von Neumann algebra. The left adjoint sends such a data of (M,X) to the free C-graded extended von Neumann algebra generated by M in degree 0 and X in degree 1.

Leopold Schlicht (May 10 2021 at 19:33):

Thanks!