Positive isometric Fourier multipliers on non-commutative $L^p$-spaces

This paper characterizes positive surjective isometric Fourier multipliers on non-commutative $L^p$-spaces of a locally compact group for $p \neq 2$, proving that such operators arise if and only if their symbols are locally almost everywhere equal to continuous characters of the group, thereby extending previous results from the unimodular to the general setting.

The Gist

Imagine you are a master architect working on a very strange, invisible city. This city isn't made of bricks and mortar, but of sounds and movements. In mathematics, this city is called a Group, and the sounds are functions that describe how things move around.

For a long time, mathematicians studied this city when it was "symmetrical" (like a perfect circle or a flat plane). In these symmetrical cities, they discovered a fascinating rule: If you have a machine that takes a sound, changes it, and outputs a new sound without changing its volume or shape (an "isometry"), that machine can only be doing one of two things:

1. It's just shifting the sound to a different location (like sliding a song forward in time).
2. It's multiplying the sound by a specific, steady tone (a "character").

This paper by Krieglir, Le Merdy, and Zadeh asks: What happens if the city is lopsided?

The Problem: The Lopsided City

In the real world, many groups (cities) are not symmetrical. Think of the "ax + b" group, which models things like zooming in and out while moving. In these cities, the rules of "volume" and "distance" are tricky. The left side of the city behaves differently than the right side. This is called a non-unimodular setting.

In these lopsided cities, the usual tools for measuring volume (called a "trace") break down. It's like trying to measure the weight of a cloud using a scale designed for rocks; the scale doesn't work right. Because of this, mathematicians couldn't prove if the "rigid rule" (that only shifts and steady tones work) still held true.

The Solution: A New Way to Measure

The authors built a new, specialized measuring tape (using something called the Connes-Hilsum construction) that works even in these lopsided, non-symmetrical cities.

They focused on a specific type of machine called a Fourier Multiplier.

• The Metaphor: Imagine a Fourier Multiplier as a sound filter. You feed it a complex song (a function), and it tweaks the frequencies.
• The Goal: They wanted to know: If this filter is a "perfect" machine (it preserves the exact shape and size of every song it touches), what kind of filter is it?

The Big Discovery

The authors proved that even in these messy, lopsided cities, the rule still holds!

If you have a filter that:

1. Preserves the shape (Isometry),
2. Preserves the "positive" nature of the sound (Positive), and
3. Can create any possible sound (Surjective/onto),

Then, that filter must be a simple, steady tone (a continuous character) that just shifts the sound around. It cannot be a complex, weird filter.

In simple terms: Even in a chaotic, asymmetrical world, the only way to perfectly preserve the "shape" of a mathematical sound is to use the simplest, most fundamental building blocks. Complexity cannot masquerade as perfect preservation.

Why Does This Matter?

1. Rigidity: It shows that mathematical structures are incredibly rigid. You can't sneak in a complex, weird operation and pretend it's a perfect transformation. Nature (or math) forces you to be simple if you want to be perfect.
2. New Tools: The authors had to invent new mathematical "gears" to handle the lopsidedness. These tools can now be used by other mathematicians to solve different problems in non-symmetrical environments.
3. Completing the Puzzle: They finished a puzzle that was started by previous mathematicians who only looked at the symmetrical parts of the city. Now, we know the rule applies to the whole city, no matter how twisted it gets.

The Takeaway

Think of it like a dance. In a perfectly symmetrical ballroom, you know that the only way to move a dancer without changing their pose is to either slide them across the floor or spin them in place. This paper proves that even if the ballroom is tilted, the floor is slippery, and the walls are curved (a non-unimodular group), those are still the only two moves allowed if you want to keep the dancer's pose perfectly intact. Any other move would distort the dance.

Technical Summary

Here is a detailed technical summary of the paper "Positive Isometric Fourier Multipliers on Non-Commutative $L^p$-Spaces" by Christoph Kriegler, Christian Le Merdy, and Safoura Zadeh.

1. Problem Statement and Context

The Core Problem:
The paper investigates the structure of Fourier multipliers acting on non-commutative $L^p$-spaces associated with a locally compact group $G$, denoted as $L^p(LG)$. Specifically, the authors aim to characterize the symbols $\phi \in L^\infty(G)$ for which the associated Fourier multiplier operator $M_{\phi, p}$ is a positive surjective isometry.

Background and Motivation:

• Commutative Case: For abelian groups $\Gamma$, Parrott and Strichartz established that for $p \neq 2$, a Fourier multiplier on $L^p(\Gamma)$ is an isometry if and only if its symbol is a continuous character of the dual group (multiplied by a constant of modulus 1). This reveals a strong rigidity phenomenon.
• Non-Commutative/Unimodular Case: Recent work by the authors (with C. Arhancet) extended this characterization to the unimodular setting (where the modular function $\Delta \equiv 1$), where the Plancherel weight is a trace.
• The Gap: The paper addresses the non-unimodular setting. In this case, the modular function $\Delta$ introduces an intrinsic asymmetry between left and right structures, and the Plancherel weight is no longer a trace. This absence of a trace creates significant analytical difficulties, rendering previous techniques (relying on trace properties) inapplicable. The goal is to extend the rigidity characterization to this general, non-tracial setting.

2. Methodology and Framework

The authors utilize the Connes–Hilsum construction of non-commutative $L^p$-spaces, which is well-suited for non-tracial weights.

Key Technical Tools:

1. Connes–Hilsum Spaces: They define $L^p(LG)$ using the spatial derivative of the left Plancherel weight $\phi_0$ with respect to the right Plancherel weight $\psi_0$. A crucial identity used is $\frac{d\phi_0}{d\psi_0} = \Delta$, where $\Delta$ is the modular function.
2. Dense Subspaces: The analysis relies on dense subspaces of $L^p(LG)$ constructed from compactly supported functions $H \subset L^2(G)$. Specifically, they study operators of the form $\Delta^{\frac{1}{2p}}\lambda(f)\Delta^{\frac{1}{2p}}$ where $f \in H \star H$.
3. Haagerup–Junge–Xu Extension Theorem: A central methodological pillar is the use of this theorem to extend positive, contractive maps from the von Neumann algebra level ($L^\infty(LG)$) to the $L^p$ level. This allows them to transfer properties like positivity and contractivity across different $p$ values.
4. Jordan Homomorphisms: To characterize isometries, the authors employ Sherman's theorem on isometries of non-commutative $L^p$-spaces. This theorem states that any surjective positive isometry between $L^p$-spaces ($p \neq 2$) is induced by a Jordan isomorphism of the underlying von Neumann algebras.
5. Approximate Units and Limits: The proof of the main characterization involves constructing specific nets of approximate units in $L^1(G)$ and analyzing the Strong Operator Topology (SOT) limits of the multiplier action on these units to deduce pointwise properties of the symbol $\phi$.

3. Key Contributions and Results

A. Equivalence of Definitions (Section 3)

The authors first establish that various natural formulations of bounded Fourier multipliers on $L^p(LG)$ are equivalent. They define a bounded Fourier multiplier $\phi$ via the inequality:
$$\|\Delta^{\frac{1}{2p}}\lambda(\phi f)\Delta^{\frac{1}{2p}}\|_p \leq C \|\Delta^{\frac{1}{2p}}\lambda(f)\Delta^{\frac{1}{2p}}\|_p$$
for $f$ in a dense subspace. They prove this definition is robust regardless of the specific placement of the modular function powers (parameterized by $\theta \in [0,1]$) and establish duality results relating $L^p$ and $L^{p'}$ multipliers.

B. Extension of Positivity (Section 4)

Using the Haagerup–Junge–Xu theorem, they prove that if a symbol $\phi$ induces a positive map on the group von Neumann algebra $LG$, it induces a positive map on $L^p(LG)$ for all $1 \leq p < \infty$.

• Corollary 4.3: If $\phi$ is a continuous character of $G$, then $M_{\phi, p}$ is a positive, surjective isometry on $L^p(LG)$ for all $p$. This establishes the "sufficient" condition.

C. Main Characterization Theorem (Section 5)

The central result of the paper is Theorem 5.3:

Let $G$ be a locally compact group, $1 < p \neq 2 < \infty$, and$\phi \in L^\infty(G)$. If the Fourier multiplier$M_{\phi, p}$is a **positive surjective isometry** on$L^p(LG)$, then$\phi$coincides locally almost everywhere with a **continuous character** of$G$.

Proof Sketch:

1. Assume $M_{\phi, p}$ is a positive surjective isometry.
2. By Sherman's theorem, this implies the existence of a Jordan isomorphism $J: LG \to LG$ satisfying a specific commutation relation with the multiplier.
3. By testing this relation against carefully chosen approximate units and taking SOT limits, the authors derive the identity $\phi(e)J(\lambda(s)) = \phi(s)\lambda(s)$.
4. This identity forces $\phi$ to be multiplicative (a character) up to a constant. Since the map is an isometry, the constant must be 1.

D. Endpoint Cases (Section 6)

• Case $p=1$: Proposition 6.1 shows that the result holds for $p=1$ even without the positivity assumption. If $M_{\phi, 1}$ is a surjective isometry, $\phi$ is a character (up to a unimodular constant). This relies on the non-commutative Banach-Stone theorem (Kadison).
• Case $p=2$: The situation is more subtle. Theorem 6.5 proves that if $M_{\phi, 2}$ is a 2-positive surjective isometry, then $\phi$ is a character. The authors note that the full extension of the $p \neq 2$ result to $p=2$ without the 2-positivity assumption remains an open question.

E. Surjectivity for $p \geq 2$

Proposition 5.4 demonstrates that for $p \geq 2$, if $M_{\phi, p}$ is an isometry, it is automatically surjective. This simplifies the hypothesis for the main theorem in the range $p \geq 2$.

4. Significance and Impact

1. Resolution of the Non-Unimodular Case: This paper successfully removes the unimodularity restriction from the characterization of isometric Fourier multipliers. It demonstrates that the rigidity phenomenon (isometries are essentially characters) is intrinsic to the group structure and holds even in the presence of the modular function's asymmetry.
2. Technical Advancement: The work provides a robust framework for handling non-tracial $L^p$-spaces. By successfully navigating the difficulties posed by the modular function $\Delta$ and the lack of a trace, the authors develop new techniques (specifically the interplay between Connes-Hilsum spaces and Jordan homomorphisms) that are likely applicable to other problems in non-commutative harmonic analysis.
3. Completeness of the Theory: By covering the full range $1 \leq p \leq \infty$(with specific conditions for$p=2$) and establishing duality and extension properties, the paper provides a comprehensive theory of Fourier multipliers on group von Neumann algebras.
4. Connection to Operator Algebras: The results deepen the understanding of the relationship between the geometric structure of the group (characters) and the operator-theoretic structure of its associated $L^p$-spaces, reinforcing the link between harmonic analysis and the theory of von Neumann algebras.

In summary, the paper establishes that the "rigidity" of isometric Fourier multipliers is a universal feature of locally compact groups, regardless of whether they are unimodular or not, provided the multiplier is positive (or the space is $L^1$).