arXiv subjects

Charles Duquet

Publications and source records attributed to Charles Duquet.

4 recordsLinked to original sources

arXiv2025

A characterization of absolutely dilatable Schur multipliers

Let $M$ be a von Neumann algebra equipped with a normal semi-finite faithful trace (nsf trace in short) and let $T\colon M\to M$ be a contraction. We say that $T$ is absolutely dilatable if there exist another von Neumann algebra $M'$ equipped with a nsf trace, a $w^*$-continuous trace preserving unital $*$-homomorphim $J\colon M\to M'$ and a trace preserving $*$-automomorphim $U\colon M'\to M'$ such that $T^k=E U^k J$ for all integer $k\geq 0$, where $E\colon M'\to M$ is the conditional expectation associated with $J$. Given a $σ$-finite measure space $(Ω,μ)$, we characterize bounded Schur multipliers $ϕ\in L^\infty(Ω^2)$ such that the Schur multiplication operator $T_ϕ\colon B(L^2(Ω))\to B(L^2(Ω))$ is absolutely dilatable. In the separable case, they are characterized by the existence of a von Neumann algebra $N$ with a separable predual, equipped with a normalized normal faithful trace $τ_N$, and of a $w^*$-continuous essentially bounded function $d\colonΩ\to N$ such that $ϕ(s,t)=τ_N(d(s)^*d(t))$ for almost every $(s,t)\inΩ^2$.

math.OA↗

arXiv2025

Absolute dilations of ucp self-adjoint Fourier multipliers: the non unimodular case

Let $φ$ be a normal semi-finite faithful weight on a von Neumann algebra $A$,let $(σ^φ_r)_{r\in{\mathbb R}}$ denote the modular automorphism group of $φ$, and let $T\colon A\to A$ be a linear map. We say that $T$ admits an absolute dilation if there exist another von Neumann algebra $M$ equipped with a normal semi-finite faithful weight $ψ$, a $w^*$-continuous, unital and weight-preserving $*$-homomorphism $J\colon A\to M$ such that $σ^ψ\circ J=J\circ σ^φ$, as well as a weight-preserving $*$-automorphism $U\colon M\to M$ such that $T^k={\mathbb E}_JU^kJ$ for all integer $k\geq 0$, where ${\mathbb E}_J\colon M\to A$ is the conditional expectation associated with $J$. Given any locally compact group $G$ and any real valued function $u\in C_b(G)$, we prove that if $u$ induces a unital completely positive Fourier multiplier $M_u\colon VN(G) \to VN(G)$, then $M_u$ admits an absolute dilation. Here $VN(G)$ is equiped with its Plangherel weight $φ_G$. This result had been settled by the first named author in the case when $G$ is unimodular so the salient point in this paper is that $G$ may be non unimodular, and hence $φ_G$ may not be a trace. The absolute dilation of $M_u$ implies that for any $1<p<\infty$, the $L^p$-realization of $M_u$ can be dilated into an isometry acting on a non-commutative $L^p$-space. We further prove that if $u$ is valued in $[0,1]$, then the $L^p$-realization of $M_u$ is a Ritt operator with a bounded $H^\infty$-functional calculus.

math.OA↗

arXiv2023

Unital positive Schur multipliers on $S_n^p$ with a completely isometric dilation

Let $1<p\not=2<\infty$ and let $S^p_n$ be the associated Schatten von Neumann class over $n\times n$ matrices. We prove new characterizations of unital positive Schur multipliers $S^p_n\to S^p_n$ which can be dilated into an invertible complete isometry acting on a non-commutative $L^p$-space. Then we investigate the infinite dimensional case.

math.FA↗

arXiv2022

Dilation properties of measurable Schur multipliers and Fourier multipliers

In the article, we find new dilatation results on non-commutative $L_p$ spaces. We prove that any selfadjoint, unital, positive measurable Schur multiplier on some $B(L^2(Σ))$ admits, for all $1\leq p<\infty$, an invertible isometric dilation on some non-commutative $L^p$-space. We obtain a similar result for selfadjoint, unital, completely positive Fourier multiplier on $VN(G)$, when $G$ is a unimodular locally compact group. Furthermore, we establish multivariable versions of these results.

math.FA↗