arXiv subjects

Cédric Arhancet

Publications and source records attributed to Cédric Arhancet.

At least 19 recordsLinked to original sources

arXiv2026

A reverse Riesz estimate combined with a spectral gap implies a Poincaré inequality

Working at the level of a sectorial operator $A$ on a Banach space $X$ and an unbounded operator $\partial$ defined on a subspace of $X$ in another Banach space $Y$, we show that a single reverse Riesz estimate $\|A^αx\|_X \lesssim \|\partial x\|_Y$ for some $0 < α< 1$, combined with the condition $0 \in ρ(A_0)$, where $A_0$ is the part of $A$ on the closure of the range of $A$, implies the Poincaré inequality $\|x - P(x)\|_X \lesssim \|\partial x\|_Y$, where $P$ is the Abel-ergodic projection onto the kernel of $A$. The condition $0 \in ρ(A_0)$ is the natural abstract substitute for a spectral gap, and is sharp already in the Hilbertian case. We also obtain a companion divergence inequality. The arguments are remarkably short, yet the principle is genuinely unifying: it covers commutative and noncommutative situations on the same footing and can be used with arbitrary Banach spaces. As a consequence, we recover Poincaré inequalities associated with hypercontractive semigroups and extend their validity to semigroups satisfying only a reduced spectral gap and a reverse Riesz estimate. We then illustrate the flexibility of the method across a wide spectrum of geometries, ranging from Riemannian manifolds, Lie groups, metric measure spaces, spin manifolds to genuinely noncommutative settings such as group von Neumann algebras, semigroups of Schur multipliers, $q$-Ornstein-Uhlenbeck semigroups and quantum tori, where we sometimes establish new inequalities and otherwise recover classical ones from a single principle.

math.FA↗

arXiv2026

A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces

We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}^2$ plurisubharmonic function $Φ$, then, at any point of its range, the Levi form of $Φ$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}^4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}^4$-space associated with a $\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}^*$-$\mathrm{TRO}$.

math.FA↗

arXiv2026

Spectral nonassociative $\mathrm{L}^p$-spaces for $\mathrm{JBW}^*$-algebras

We complete the construction of tracial spectral nonassociative $\mathrm{L}^p$-spaces for general $\mathrm{JBW}^*$-algebras. More precisely, if $\mathcal{M}$ is a $\mathrm{JBW}^*$-algebra equipped with a normal finite faithful trace $τ$ and $1 \leq p < \infty$, we prove that $\|x\|_{\mathrm{L}^p(\mathcal{M})} \overset{\mathrm{def}}{=} (τ[(x^* \circ x)^{\frac p2}])^{\frac1p}$, where $x \in \mathcal{M}$, defines a norm on $\mathcal{M}$. This resolves the remaining exceptional case left open by the corresponding result for $\mathrm{JW}^*$-algebras. The main difficulty is the complexified Albert algebra $\mathrm{H}_3(\mathbb{O}_{\mathbb{C}})$, which admits no embedding into an associative operator algebra. To treat this case, we establish a Jordan analogue of the joint convexity of the Kiefer map $\mathrm{M}_n \times \mathrm{H}_n^{++} \to \mathrm{H}_n^{+}$, $(a,h) \mapsto a^*h^{-1}a$, where $\mathrm{H}_n$ is the space of Hermitian matrices, and a Jordan version of a variational formula of Carlen and Lieb. This provides a complex Banach space framework for Jordan-algebraic models arising in some generalized probabilistic theories.

math.FA↗

arXiv2026

Multiplicativity of the weak Haagerup constant

We prove that the weak Haagerup constant is multiplicative with respect to direct products, namely $Λ_{\mathrm{WH}}(G \times H)=Λ_{\mathrm{WH}}(G) Λ_{\mathrm{WH}}(H)$ for any locally compact groups $G$ and $H$. Knudby previously proved the submultiplicative inequality and asked whether equality always holds.

math.FA↗

arXiv2026

Optimal square size for separating the operator and completely bounded norms of Schur multipliers on $S^4$

We determine the smallest square size at which the operator norm and the completely bounded norm of a Schur multiplier on $S^4$ can differ. More precisely, for $A=\begin{bmatrix}1&2&0\\1&-2&0\\2&2\mathrm{i}&0\end{bmatrix}$ we prove \[ \|M_A\|_{S^4_3\to S^4_3}^2\leq\frac{1119}{250} <\left(\frac{18983532}{947485}\right)^{1/2} \leq \|\mathrm{Id}_{S^4_2}\otimes M_A\|_{S^4_6\to S^4_6}^2. \] We also prove that no $2\times2$ Schur multiplier can have unequal operator and completely bounded norms on $S^4$, so square size three is optimal. In addition, we determine the exact ordinary norm of the previous multiplier analytically and give an independent computer-assisted certification of a rational upper bound.

math.FA↗

arXiv2026

Completely isometric subspaces of noncommutative $\mathrm{L}^p$-spaces and contractive projections

We investigate the relation between the complete isometry class of subspaces of noncommutative $\mathrm{L}^p$-spaces and their contractive complementability, where $1 < p < \infty$ with $p \not= 2$. We show that if $P \colon \mathrm{L}^p(\mathcal{M}) \to \mathrm{L}^p(\mathcal{M})$ is a positive contractive projection whose range is completely isometric to another noncommutative $\mathrm{L}^p$-space, then $P$ is necessarily completely positive. This provides a converse to the main result of [ArR24] and the positivity assumption on $P$ is essential. We further establish a rectangular analogue of this result. More precisely, we prove that every closed subspace of a noncommutative $\mathrm{L}^p$-space which is completely isometric to a rectangular noncommutative $\mathrm{L}^p$-space of the form $e\mathrm{L}^p(\mathcal{N})(1-e)$ is the range of a contractively decomposable projection. Combined with the known converse implication, this yields a characterization of the ranges of contractively decomposable projections as precisely the subspaces completely isometric to rectangular $\mathrm{L}^p$-spaces associated with $\mathrm{W}^*$-ternary rings of operators.

math.FA↗

arXiv2026

On the convex structure of the space of quantum channels which act as Fourier multipliers

If $G$ is a compact group, continuous normalized positive definite functions are in one-to-one correspondence with unital quantum channels acting as Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$. We study the convex geometry of the convex set $\mathrm{P}_1(G)$ of normalized positive definite functions, equipped with the topology induced by the norm topology of the Fourier algebra $\mathrm{A}(G)$, and its relation with the structure of $\mathrm{VN}(G)$. We show that the von Neumann algebras of two compact groups $G$ and $H$ are $*$-isomorphic if and only if the convex sets $\mathrm{P}_1(G)$ and $\mathrm{P}_1(H)$ are affinely homeomorphic. We also describe the group of affine homeomorphisms of $\mathrm{P}_1(G)$ in terms of Jordan $*$-automorphisms of $\mathrm{VN}(G)$.

math.OA↗

arXiv2026

Curvature, Dolbeault-Dirac operators, and an $\mathrm{L}^p$-index theorem on compact Kähler manifolds

We develop an $\mathrm{L}^p$-Banach noncommutative-geometric framework for Dolbeault-Dirac operators on compact Kähler manifolds with coefficients in a Hermitian holomorphic vector bundle $E$. For every $p \in (1,\infty)$, we prove that the closed $\mathrm{L}^p$-realization $\mathcal{D}_{E,p}$ of the Dolbeault-Dirac operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on $\mathrm{L}^p(Ω^{0,\bullet}(M,E))$. We also show an $\mathrm{L}^p$-Gaffney-type estimate, obtain $\mathrm{L}^p$-Hodge decompositions, and prove that $\mathcal{D}_{E,p}$ gives rise to an even compact Banach spectral triple over the algebra $\mathrm{C}(M)$, graded by form parity. The index of the associated Fredholm operator is equal to the holomorphic Euler characteristic $χ(M,E)$. In particular, it is independent of $p$. A central tool is an abstract notion of Ricci curvature lower bound for strongly continuous semigroups on $\mathrm{UMD}$ Banach spaces, formulated as a semigroup-level intertwining relation. Under this condition, together with natural Riesz equivalences and bounded $\mathrm{H}^\infty$ functional calculi for the relevant generators, the associated Hodge-Dirac operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus. The framework also applies to heat semigroups on Riemannian manifolds, $q$-Ornstein-Uhlenbeck semigroups and semigroups of Schur multipliers. This provides a unified Banach-space approach to curvature, functional calculus, Riesz transforms and index theory beyond the Hilbert space setting.

math.FA↗

arXiv2026

The $\mathrm{L}^p$-index of the Hodge-Dirac operator on compact Riemannian manifolds

We investigate the spectral and index-theoretic properties of the Hodge-Dirac operator $D = \mathrm{d} + \mathrm{d}^*$ acting on the Banach space $\mathrm{L}^p(Ω^\bullet(M))$ of differential forms over a compact Riemannian manifold $M$. Relying on the compactness of $M$, we establish that this operator is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus, without curvature assumptions. This result enables us to prove that the triple $(\mathrm{C}(M), \mathrm{L}^p(Ω^\bullet(M)), D)$ constitutes a compact Banach spectral triple. We then investigate consistent pairings between the Banach K-homology and the K-theory of the algebra $\mathrm{C}(M)$, identifying the resulting Fredholm indices with classical topological invariants, and hence showing that they are independent of $p$. We recover the classical Euler characteristic and the Hirzebruch signature as $\mathrm{L}^p$-indices, demonstrating the effectiveness of Banach noncommutative geometry for geometric analysis, beyond the Hilbertian setting.

math.FA↗

arXiv2026

Dolbeault-Dirac operators on compact Kähler manifolds in Banach noncommutative geometry

We develop an $\mathrm{L}^p$-theory for Dolbeault-Dirac operators on compact Kähler manifolds with coefficients in a Hermitian holomorphic vector bundle $E$. For each $p \in (1,\infty)$ we consider the closed $\mathrm{L}^p$-realization $\mathcal{D}_{E,p}$ of the Dolbeault-Dirac operator $\mathcal{D}_{E}$ on the Banach space $\mathrm{L}^p(Ω^{0,\bullet}(M,E))$. We prove that $\mathcal{D}_{E,p}$ is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus. We establish a Gaffney-type estimate controlling covariant derivatives in $\mathrm{L}^p$, and also obtain $\mathrm{L}^p$-Hodge decompositions. As an application, we show that the closed operator $\mathcal{D}_{E,p}$ yields a compact Banach spectral triple, and we identify the index of the associated Fredholm operator with the holomorphic Euler characteristic, proving in particular that it is independent of $p$. This work initiates a connection between complex geometry, $\mathrm{L}^p$-analysis and Banach noncommutative geometry, beyond the Hilbert space setting.

math.FA↗

arXiv2026

Spectral versus interpolation norms in tracial nonassociative $\mathrm{L}^p$-spaces

We investigate the metric structure of nonassociative $\mathrm{L}^p$-spaces associated with tracial $\mathrm{JW}^*$-algebras. While noncommutative $\mathrm{L}^p$-spaces arising from von Neumann algebras enjoy a unique natural norm, the situation in the Jordan setting is more subtle. We compare two canonical definitions: the interpolation norm, arising from the complex method between the algebra and its predual, and the spectral norm, defined with the trace. We show that these two norms are equivalent but generally not isometric for $p \neq 2$, even in the associative case of nonabelian von Neumann algebras when viewed through the Jordan product, thereby answering an open question raised by the first author in a previous paper. We further analyze the geometry of these spaces in concrete examples as complex spin factors or the complexified Albert algebra. Finally, we discuss the relevance of these results to generalized probabilistic theories (GPTs), where Jordan structures arise naturally, and explain why $\mathrm{JBW}$-algebras and their preduals provide a natural framework for such models.

math.OA↗

arXiv2026

Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields

For any non-Archimedean local field $\mathbb{K}$ and any integer $n \geq 1$, we show that the Taibleson operator admits a bounded $\mathrm{H}^\infty(Σ_θ)$ functional calculus on the Bochner space $\mathrm{L}^p(\mathbb{K}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $θ> 0$, where $Σ_θ=\{ z \in \mathbb{C}^*: |\arg z| < θ\}$ and $1 < p < \infty$. Moreover, we prove that it even admits a bounded Hörmander functional calculus of order $\frac{3}{2}$. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the $R$-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued $\mathrm{L}^p$-spaces over totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator.

math.CA↗

arXiv2025

Classical harmonic analysis viewed through the prism of noncommutative geometry

The aim of this paper is to bridge noncommutative geometry with classical harmonic analysis on Banach spaces, focusing primarily on both classical and noncommutative $\mathrm{L}^p$ spaces. Introducing a notion of Banach Fredholm module, we define new abelian groups, $\mathrm{K}^{0}(\mathcal{A},\mathscr{B})$ and $\mathrm{K}^{1}(\mathcal{A},\mathscr{B})$, of $\mathrm{K}$-homology associated with an algebra $\mathcal{A}$ and a suitable class $\mathscr{B}$ of Banach spaces, such as the class of $\mathrm{L}^p$-spaces. We establish index pairings of these groups with the $\mathrm{K}$-theory groups of the algebra $\mathcal{A}$. Subsequently, by considering (noncommutative) Hardy spaces, we uncover the natural emergence of Hilbert transforms, leading to Banach Fredholm modules and culminating in new index theorems. Moreover, by associating each reasonable sub-Markovian semigroup of operators with a < >, we explain how this leads to (possibly kernel-degenerate) Banach Fredholm modules, thereby revealing the role of vectorial Riesz transforms in this context. Overall, our approach significantly integrates the analysis of operators on $\mathrm{L}^p$-spaces into the expansive framework of noncommutative geometry, offering new perspectives.

math.FA↗

arXiv2025

The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples

This paper investigates the functional calculus of the harmonic oscillator on each Moyal-Groenewold plane, the noncommutative phase space which is a fundamental object in quantum mechanics. Specifically, we show that the harmonic oscillator admits a bounded $\mathrm{H}^\infty(Σ_ω)$ functional calculus for any angle $0 < ω< \fracπ{2}$ and even a bounded Hörmander functional calculus on the associated noncommutative $\mathrm{L}^p$-spaces, where $Σ_ω=\{ z \in \mathbb{C}^*: |\arg z| <ω\}$. To achieve these results, we develop a connection with the theory of 2-step nilpotent Lie groups by introducing a notion of twisted Weyl tuple and connecting it to some semigroups of operators previously investigated by Robinson via group representations. Along the way, we demonstrate that $\mathrm{L}^p$-square-max decompositions lead to new insights between noncommutative ergodic theory and $R$-boundedness, and we prove a twisted transference principle, which is of independent interest. Our approach accommodates the presence of a constant magnetic field and they are indeed new even in the framework of magnetic Weyl calculus on classical $\mathrm{L}^p$-spaces. Our results contribute to the understanding of functional calculi on noncommutative spaces and have implications for the maximal regularity of the most basic evolution equations associated to the harmonic oscillator.

math.FA↗

arXiv2025

Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability

We study the algebra $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$, and its relation to the Fourier-Stieltjes algebra $\mathrm{B}(G)$. For discrete groups, we prove that these two algebras coincide isometrically. In contrast, we show that the identity $\mathfrak{M}^{\infty,\mathrm{dec}}(G) = \mathrm{B}(G)$ fails for various classes of non-discrete groups, and that, among second-countable unimodular groups, inner amenability ensures the equality. Our approach relies on the existence of contractive projections preserving complete positivity from the space of completely bounded weak* continuous operators on $\mathrm{VN}(G)$ onto the subspace of completely bounded Fourier multipliers. We show that such projections exist in the inner amenable case. As an application, we obtain a new operator-algebraic characterization of amenability. We also investigate the analogous problem for the space of completely bounded Fourier multipliers on the noncommutative $\mathrm{L}^p$-spaces $\mathrm{L}^p(\mathrm{VN}(G))$, for $1 \leq p \leq \infty$. Using Lie group theory and results stemming from the solution to Hilbert's fifth problem, we prove that second-countable unimodular finite-dimensional amenable locally compact groups admit compatible projections at $p = 1$ and $p = \infty$. These results reveal new structural links between harmonic analysis, operator algebras, and the geometry of locally compact groups.

math.FA↗

arXiv2025

Quantum information theory and Fourier multipliers on quantum groups

In this paper, we compute the exact values of the minimum output entropy and the completely bounded minimal entropy of very large classes of quantum channels acting on matrix algebras $\mathrm{M}_n$. Our new and simple approach relies on the theory of locally compact quantum groups and our results use a new and precise description of bounded Fourier multipliers from $\mathrm{L}^1(\mathbb{G})$ into $\mathrm{L}^p(\mathbb{G})$ for $1 < p \leq \infty$ where $\mathbb{G}$ is a co-amenable locally compact quantum group and on the automatic completely boundedness of these multipliers that this description entails. Indeed, our approach even allows to use convolution operators on quantum hypergroups. This enable us to connect equally the topic of computation of entropies and capacities to subfactor planar algebras. We also give a upper bound of the classical capacity of each considered quantum channel which is already sharp in the commutative case. Quite surprisingly, we observe by direct computations that some Fourier multipliers identifies to direct sums of classical examples of quantum channels (as dephasing channel or depolarizing channels). Indeed, we show that the study of unital qubit channels can be seen as a part of the theory of Fourier multipliers on the von Neumann algebra of the quaternion group $\mathbb{Q}_8$. Unexpectedly, we also connect ergodic actions of (quantum) groups to this topic of computation, allowing some transference to other channels. We also connect the Quantum Harmonic analysis of Werner. Finally, we investigate entangling breaking and $\mathrm{PPT}$ Fourier multipliers and we characterize conditional expectations which are entangling breaking.

math.OA↗

arXiv2024

Entanglement-assisted classical capacities of some channels acting as radial multipliers on fermion algebras

We investigate a new class of unital quantum channels on $\mathrm{M}_{2^k}$, acting as radial multipliers when we identify the matrix algebra $\mathrm{M}_{2^k}$ with a finite-dimensional fermion algebra. Our primary contribution lies in the precise computation of the (optimal) rate at which classical information can be transmitted through these channels from a sender to a receiver when they share an unlimited amount of entanglement. Our approach relies on new connections between fermion algebras with the $n$-dimensional discrete hypercube $\{-1,1\}^n$. Significantly, our calculations yield exact values applicable to the operators of the fermionic Ornstein-Uhlenbeck semigroup. This advancement not only provides deeper insights into the structure and behaviour of these channels but also enhances our understanding of Quantum Information Theory in a dimension-independent context.

quant-ph↗

arXiv2024

Spectral triples, Coulhon-Varopoulos dimension and heat kernel estimates

We investigate the relations between the (completely bounded) local Coulhon-Varopoulos dimension and the spectral dimension of spectral triples associated to sub-Markovian semigroups (or Dirichlet forms) acting on classical (or noncommutative) $\mathrm{L}^p$-spaces associated to finite measure spaces. More precisely, we prove that the completely bounded local Coulhon-Varopoulos dimension $d$ exceeds the spectral dimension and even implies that the associated Hodge-Dirac operator is $d^+$-summable. We explore different settings to compare these two values: compact Riemannian manifolds, compact Lie groups, sublaplacians, metric measure spaces, noncommutative tori and quantum groups. Specifically, we prove that, while very often equal in smooth compact settings, these dimensions can diverge. Finally, we show that the existence of a symmetric sub-Markovian semigroup on a von Neumann algebra with finite completely bounded local Coulhon-Varopoulos dimension implies that the von Neumann algebra is necessarily injective.

math.OA↗