Searcharxiv⌕ Search

arXiv subjects

Cédric Arhancet

Publications and source records attributed to Cédric Arhancet.

At least 37 records · Page 2Linked to original sources
arXiv2024

2-positive contractive projections on noncommutative $\mathrm{L}^p$-spaces

We prove the first theorem on projections on general noncommutative $\mathrm{L}^p$-spaces associated with non-type I von Neumann algebras where $1 \leqslant p < \infty$. This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative $\mathrm{L}^p$-space is completely order isometrically isomorphic to some noncommutative $\mathrm{L}^p$-space. This result is sharp and is even new for Schatten spaces $S^p$. Our approach relies on non-tracial Haagerup's noncommutative $\mathrm{L}^p$-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman.

math.OA↗
arXiv2024

Nonassociative $\mathrm{L}^p$-spaces and embeddings in noncommutative $\mathrm{L}^p$-spaces

We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $φ$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More precisely, we show that our nonassociative $\mathrm{L}^p$-spaces contain isometrically the $\mathrm{L}^p$-spaces of Iochum and that all tracial nonassociative $\mathrm{L}^p$-spaces from $\mathrm{JW}^*$-factors arise as positively contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces.

math.OA↗
arXiv2023

Contractively decomposable projections on noncommutative $\mathrm{L}^p$-spaces

We describe and characterize the contractively decomposable projections on noncommutative $\mathrm{L}^p$-spaces. Our result relies on a new lifting result for decomposable maps of independent interest and on some tools from ergodic theory. Our theorem is new even for finite-dimensional Schatten spaces. Our description allows us to connect this topic with $\mathrm{W}^*$-ternary rings of operators and a slight generalization of our result for more general projections makes $\mathrm{JBW}^*$-triples appear in this context. We also prove that all rectangular $\mathrm{L}^p$-spaces associated with $\mathrm{W}^*$-ternary rings of operators arise as contractively decomposable complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. Finally, we introduce a notion of $\mathrm{L}^p$-space associated to each $σ$-finite $\mathrm{JBW}^*$-triple and we explain the link with the context of this paper.

math.OA↗
arXiv2023

Positive contractive projections on noncommutative $\mathrm{L}^p$-spaces and nonassociative $\mathrm{L}^p$-spaces

We continue our investigation of contractive projections on noncommutative $\mathrm{L}^p$-spaces where $1 < p < \infty$ started in \cite{ArR19}. We improve the results of \cite{ArR19} and we characterize precisely the positive contractive projections on a noncommutative $\mathrm{L}^p$-space associated with a $σ$-finite von Neumann algebra. We connect this topic to the theory of $\mathrm{JW}^*$-algebras. More precisely, in large cases, we are able to show that the range of a positive contractive projection is isometric to a nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JW}^*$-algebra.

math.OA↗
arXiv2023

Dilations of markovian semigroups of measurable Schur multipliers

Using probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm{B}(\mathrm{L}^2(X))$ of bounded operators on the Hilbert space $\mathrm{L}^2(X)$, where $X$ is a suitable measure space, can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to $\mathrm{BMO}$-spaces. We also give an answer to a question of Steen, Todorov and Turowska on completely positive continuous Schur multipliers.

math.OA↗
arXiv2023

Separating Fourier and Schur multipliers

Let $G$ be a locally compact unimodular group, let $1\leq p<\infty$,let $ϕ\in L^\infty(G)$ and assume that the Fourier multiplier $M_ϕ$associated with $ϕ$ is bounded on the noncommutative $L^p$-space $L^p(VN(G))$.Then $M_ϕ\colon L^p(VN(G))\to L^p(VN(G))$ is separating (that is,$\{a^*b=ab^*=0\}\Rightarrow\{M_ϕ(a)^* M_ϕ(b)=M_ϕ(a)M_ϕ(b)^*=0\}$for any $a,b\in L^p(VN(G))$) if and only if thereexists $c\in\mathbb C$ and a continuouscharacter $ψ\colon G\to\mathbb C$ such that $ϕ=cψ$ locally almost everywhere. This provides a characterization of isometricFourier multipliers on $L^p(VN(G))$, when $p\not=2$. Next, let $Ω$ be a $σ$-finite measure space, let $ϕ\in L^\infty(Ω^2)$and assume that the Schur multiplier associated with $ϕ$ is bounded on the Schatten space $S^p(L^2(Ω))$. We prove that this multiplier is separating if and only if there exist a constant $c\in\mathbb C$ and two unitaries $α,β\in L^\infty(Ω)$ such that $ϕ(s,t) =c\, α(s)β(t)$ a.e. on $Ω^2.$ This provides acharacterization of isometric Schur multiplierson $S^p(L^2(Ω))$, when $p\not=2$.

math.CA↗
arXiv2022

Sobolev algebras on Lie groups and noncommutative geometry

We show that there exists a quantum compact metric space which underlies the setting of each Sobolev algebra associated to a subelliptic Laplacian $Δ=-(X_1^2+\cdots+X_m^2)$ on a compact connected Lie group $G$ if $p$ is large enough, more precisely under the (sharp) condition $p > \frac{d}α$ where $d$ is the local dimension of $(G,X)$ and where $0 < α\leq 1$. We also provide locally compact variants of this result and generalizations for real second order subelliptic operators. We also introduce a compact spectral triple (=noncommutative manifold) canonically associated to each subelliptic Laplacian on a compact group. In addition, we show that its spectral dimension is equal to the local dimension of $(G,X)$. Finally, we prove that the Connes spectral pseudo-metric allows us to recover the Carnot-Carathéodory distance.

math.FA↗
arXiv2022

Markov dilations of semigroups of Fourier multipliers

We describe a Markov dilation for any weak* continuous semigroup $(T_t)_{t \geq 0}$ of selfadjoint unital completely positive Fourier multipliers acting on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$.

math.OA↗
arXiv2022

Projections, multipliers and decomposable maps on noncommutative $\mathrm{L}^p$-spaces

We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mathrm{L}^p$ operators onto the subspace of Schur or Fourier multipliers, preserving complete positivity. On the other hand, we show the existence of bounded Fourier multipliers which cannot be approximated by regular operators, on large classes of locally compact groups, including all infinite abelian locally compact groups. We finish by introducing a general procedure for proving positive results on selfadjoint contractively decomposable Fourier multipliers, beyond the amenable case.

math.OA↗
arXiv2022

Dilations of markovian semigroups of Fourier multipliers on locally compact groups

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of Markov Fourier multipliers acting on a group von Neumann algebra $\mathrm{VN}(G)$ associated to a locally compact group $G$ can be dilated by a weak* continuous group of Markov $*$-automorphisms on a bigger von Neumann algebra. Our construction relies on probabilistic tools and is even new for the group $\mathbb{R}^n$. Our results imply the boundedness of the McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these semigroups on the associated noncommutative $\mathrm{L}^p$-spaces.

math.OA↗
arXiv2020

Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I

In this work, we solve the problem explicitly stated at the end of a paper of Junge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groups including all amenable groups and free groups. More precisely, we prove that the Hodge-Dirac operator of the canonical "hidden" noncommutative geometry associated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on a bisector which implies a positive answer to the quoted problem. Our result can be seen as a strengthening of the dimension free estimates of Riesz transforms of the above authors and also allows us to provide Hodge decompositions. A part of our proof relies on a new transference argument between multipliers which is of independent interest. Our results are even new for the Poisson semigroup on $\mathbb{T}^n$. We also provide a similar result for Markov semigroups of Schur multipliers and dimension free estimates for noncommutative Riesz transforms associated with these semigroups. Along the way, we also obtain new Khintchine type equivalences for $q$-Gaussians in $\mathrm{L}^p$-spaces associated to crossed products. Our results allow us to introduce new spectral triples (i.e. noncommutative manifolds) and new quantum (locally) compact metric spaces, in connection with the carré du champ, which summarize the underlying geometry of our setting. Finally, our examples lead us to introduce a Banach space variant of the notion of spectral triple suitable for our context.

math.OA↗
arXiv2018

Dilations of semigroups on von Neumann algebras and noncommutative $\mathrm{L}^p$-spaces

We prove that any weak* continuous semigroup $(T_t)_{t \geq 0}$ of factorizable Markov maps acting on a von Neumann algebra $M$ equipped with a normal faithful state can be dilated by a group of Markov $*$-automorphisms analogous to the case of a single factorizable Markov operator, which is an optimal result. We also give a version of this result for strongly continuous semigroups of operators acting on noncommutative $\mathrm{L}^p$-spaces and examples of semigroups to which the results of this paper can be applied. Our results implies the boundedness of the McIntosh's $\mathrm{H}^\infty$ functional calculus of the generators of these semigroups on the associated noncommutative $\mathrm{L}^p$-spaces generalising some previous work from Junge, Le Merdy and Xu. Finally, we also give concrete dilations for Poisson semigroups which are even new in the case of $\mathbb{R}^n$.

math.OA↗
arXiv2018

On analyticity of semigroups on Bochner spaces and on vector-valued noncommutative $\mathrm{L}^p$-spaces

We show that the analyticity of semigroups $(T_t)_{t \geq 0}$ of (not necessarily positive) selfadjoint contractive Fourier multipliers on $\mathrm{L}^p$-spaces of any abelian locally compact group is preserved by the tensorisation of the identity operator $\mathrm{Id}_X$ of a Banach space $X$ for a large class of K-convex Banach spaces, answering partially a conjecture of Pisier. The result is even new for semigroups of Fourier multipliers acting on $\mathrm{L}^p(\mathbb{R}^n)$. The proof relies on the use of noncommutative Banach spaces and we give a more general result for semigroups of Fourier multipliers acting on noncommutative $\mathrm{L}^p$-spaces. Finally, we also give a somewhat different version of this result in the discrete case, i.e. for Ritt operators.

math.FA↗
arXiv2015

Isometric dilations and $H^\infty$ calculus for bounded analytic semigroups and Ritt operators

We show that any bounded analytic semigroup on $L^p$ (with $1<p<\infty$) whose negative generator admits a bounded $H^{\infty}$ functional calculus with respect to some angle $< π/2$ can be dilated into a bounded analytic semigroup $(R_t)_{t\geq 0}$ on a bigger $L^p$-space in such a way that $R_t$ is a positive contraction for any $t$. We also establish a discrete analogue for Ritt operators and consider the case when $L^p$-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem.

math.FA↗
arXiv2015

Analytic semigroups on vector valued noncommutative $L^p$-spaces

We give sufficient conditions on an operator space $E$ and on a semigroup of operators on a von Neumann algebra $M$ to obtain a bounded analytic or a $R$-analytic semigroup $(T_t \otimes Id_E)_{t \geq 0}$ on the vector valued noncommutative $L^p$-space $L^p(M,E)$. Moreover, we give applications to the $H^\infty(Σ_θ)$ functional calculus of the generators of these semigroups, generalizing some earlier work of M. Junge, C. Le Merdy and Q. Xu.

math.OA↗