arXiv subjects

Christian Le Merdy

Publications and source records attributed to Christian Le Merdy.

At least 19 recordsLinked to original sources

arXiv2026

A Non-commutative Individual Ergodic Theorem Along Sparse Random Subsequences

Let $(M,\tau)$ be a semifinite von Neumann algebra, let $J$ be a trace-preserving Jordan isomorphism, and let $(n_k)_{k\geq 1}$ be a random increasing sequence of integers obtained by selecting each integer $n\geq 1$ independently with probability $n^{-\alpha}$, where $0<\alpha<\frac12$. We show that, almost surely, for every $x\in L^1(M)$, $ \frac1m\sum_{k=1}^m J^{n_k}(x)$ converges bilaterally almost uniformly. This extends LaVictoire's classical random $L^1$ ergodic theorem to the non-commutative setting.

math.OA↗

arXiv2026

Connecting $H^\infty$-functional calculus and isometric dilations for commuting families of Ritt$_E$ operators

Let $(T_1,\ldots,T_d)$ be a commuting $d$-tuple of Ritt$_E$ operators on some UMD Banach space $X$. We show that $(T_1,\ldots,T_d)$ admits a bounded $H^\infty$-functional calculus if and only if $T_k$ is an $R$-Ritt$_E$ operator for every $k=1,\ldots,d$, and the $d$-tuple $(T_1,\ldots,T_d)$ admits an isometric dilation $(U_1,\ldots,U_d)$ on some UMD Banach space $Y$ such that $(U_1,\ldots,U_d)$ is polynomially bounded. In the case where $X$ further possesses property $(\alpha)$, we establish other characterizations of the $H^\infty$-functional calculus property for $(T_1,\ldots,T_d)$ in terms of isometric dilations.

math.FA↗

arXiv2026

Pointwise Convergence for Random Ergodic Averages in Non-commutative $L^p$-spaces

Let $M$ be a semifinite von Neumann algebra and $T$ a positive contraction on both $L^1(M)$ and $L^\infty(M)$. We consider ergodic averages along a random sparse subsequence determined by independent Bernoulli variables $(X_n)_{n\geq 1}$ with $\mathbb{P}(X_n = 1) = n^{-α}$, and set $W_N = \sum_{n=1}^N \mathbb{E}[X_n]$. We prove that, almost surely, the averages $\frac{1}{W_N} \sum_{n=1}^N X_n\, T^n(x)$ converge bilaterally almost uniformly to the ergodic projection for all $1 < p < \infty$. This extends a theorem of Bourgain to the non-commutative setting.

math.OA↗

arXiv2026

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

For a locally compact group $G$, let $\mathcal{L}G$ denote its left group von Neumann algebra and let $L^p(\mathcal{L}G)$, $1 \le p \le \infty$, be the corresponding non-commutative $L^p$-space. Given $ϕ\in L^\infty(G)$, we study the Fourier multiplier $M_{ϕ,p}$ acting on $L^p(\mathcal{L}G)$. We prove that for any $p \neq 2$, the operator $M_{ϕ,p}$ is a positive surjective isometry if and only if $ϕ$ coincides locally almost everywhere with a continuous character of $G$. This characterization extends results obtained recently (jointly with C.~Arhancet) in the unimodular setting.

math.OA↗

arXiv2025

Absolute dilation of Fourier multipliers

Let ${\mathcal M}$ be a von Neumann algebra equipped with a normal semifinite faithful (nsf) trace. We say that an operator $T :{\mathcal M}\to {\mathcal M}$ is absolutely dilatable if there exist another von Neumann algebra $M$ with an nsf trace, a unital normal trace preserving $\ast$-homomorphism $J: {\mathcal M} \to M$, and a trace preserving $\ast$-automorphism $U: M \to M$ such that $T^k = {\mathbb E}_J U^k J \quad \text{for all } k \geq 0,$ where ${\mathbb E}_J: M \to {\mathcal M}$ is the conditional expectation associated with $J$. For a discrete amenable group $G$ and a function $u:G\to\mathbb{C}$ inducing a unital completely positive Fourier multiplier $M_u: VN(G) \to VN(G)$, we establish the following transference theorem: the operator $M_u$ admits an absolute dilation if and only if its associated Herz-Schur multiplier does. From this result, we deduce a characterization of Fourier multipliers with an absolute dilation in this setting. Building on the transference result, we construct the first known example of a unital completely positive Fourier multiplier that does not admit an absolute dilation. This example arises in the symmetric group ${\mathcal S}_3$, the smallest group where such a phenomenon occurs. Moreover, we show that for every abelian group $G$, every Fourier multiplier always admits an absolute dilation.

math.OA↗

arXiv2025

Functional calculus for a bounded $C_0$-semigroup on Hilbert space

We introduce a new Banach algebra ${\mathcal A}({\mathbb C}_+)$ of bounded analytic functions on ${\mathbb C}_+=\{z\in{\mathbb C}\, :\, {\rm Re}(z)>0\}$ which is an analytic version of the Figa-Talamenca-Herz algebras on ${\mathbb R}$. Then we prove that the negative generator $A$ of any bounded $C_0$-semigroup on Hilbert space $H$ admits a bounded (natural) functional calculus $ρ_A\colon {\mathcal A}({\mathbb C}_+)\to B(H)$. We prove that this is an improvement of the bounded functional calculus ${\mathcal B}({\mathbb C}_+)\to B(H)$ recently devised by Batty-Gomilko-Tomilov on a certain Besov algebra ${\mathcal B}({\mathbb C}_+)$ of analytic functions on ${\mathbb C}_+$, by showing that ${\mathcal B}({\mathbb C}_+)\subset {\mathcal A}({\mathbb C}_+)$ and ${\mathcal B}({\mathbb C}_+)\not= {\mathcal A}({\mathbb C}_+)$. In the Banach space setting, we give similar results for negative generators of $γ$-bounded $C_0$-semigroups. The study of ${\mathcal A}({\mathbb C}_+)$ requires to deal with Fourier multipliers on the Hardy space $H^1({\mathbb R})\subset L^1({\mathbb R})$ of analytic functions.

math.FA↗

arXiv2025

Polygonal functional calculus for operators with finite peripheral spectrum

Let $T\colon X\to X$ be a bounded operator on Banach space, whose spectrum $σ(T)$ is included in the closed unit disc $\overline{\mathbb D}$. Assume that the peripheral spectrum $σ(T)\cap{\mathbb T}$ is finite and that $T$ satisfies a resolvent estimate $$\Vert(z-T)^{-1}\Vert\lesssim \max\bigl\{\vert z -ξ\vert^{-1}\, :\,ξ\in σ(T)\cap{\mathbb T}\bigr\}, \qquad z\in\overline{\mathbb D}^c.$$ We prove that $T$ admits a bounded polygonal functional calculus, that is, an estimate $\Vertϕ(T)\Vert\lesssim \sup\{\vertϕ(z)\vert\, :\, z\inΔ\}$ for some polygon $Δ\subset{\mathbb D}$ and all polynomials $ϕ$, in each of the following two cases : (i) either $X=L^p$ for some $1<p<\infty$, and $T\colon L^p\to L^p$ is a positive contraction; (ii) or $T$ is polynomially bounded and for all $ξ\in σ(T)\cap{\mathbb T},$ there exists a neighborhood $\mathcal V$ of $ξ$ such that the set $\{(ξ-z)(z-T)^{-1}\, :\, z\in{\mathcal V}\cap \overline{\mathbb D}^c\}$ is $R$-bounded (here $X$ is arbitrary). Each of these two results extends a theorem of de Laubenfels concerning polygonal functional calculus on Hilbert space. Our investigations require the introduction, for any finite set $E\subset{\mathbb T}$, of a notion of Ritt$_E$ operator which generalises the classical notion of Ritt operator. We study these Ritt$_E$ operators and their natural functional calculus.

math.FA↗

arXiv2025

$S^1$-bounded Fourier multipliers on $H^1({\mathbb R})$ and functional calculus for semigroups

Let $T\colon H^1({\mathbb R})\to H^1({\mathbb R})$ be a bounded Fourier multiplier on the analytic Hardy space $H^1({\mathbb R})\subset L^1({\mathbb R})$ and let $m\in L^\infty({\mathbb R}_+)$ be its symbol, that is, $\widehat{T(h)}=m\widehat{h}$ for all $h\in H^1({\mathbb R})$.Let $S^1$ be the Banach space of all trace class operators on $\ell^2$. We show that $T$ admits a bounded tensor extension $T\overline{\otimes} I_{S_1}\colon H^1({\mathbb R};S^1) \to H^1({\mathbb R};S^1)$ if and only if there exist a Hilbert space $\mathcal H$ and two functions $α, β\in L^\infty({\mathbb R}_+;{\mathcal H})$ such that $m(s+t) = \langleα(t),β(s)\rangle_{\mathcal H}$ for almost every $(s,t)\in{\mathbb R}_+^2$. Such Fourier multipliers arecalled $S^1$-bounded and we let ${\mathcal M}_{S^1}(H^1({\mathbb R}))$ denote the Banach space of all $S^1$-bounded Fourier multipliers. Next we apply this result to functional calculus estimates, in two steps. First we introduce a new Banach algebra ${\mathcal A}_{0,S^1}({\mathbb C}_+)$ of bounded analytic functions on ${\mathbb C}_+ =\bigl\{z\in{\mathbb C}\, :\, {\rm Re}(z)>0\bigr\}$ and show that its dual space coincides with ${\mathcal M}_{S^1}(H^1({\mathbb R}))$. Second, given any bounded $C_0$-semigroup $(T_t)_{t\geq 0}$ on Hilbert space, and any $b\in L^1({\mathbb R}_+)$, we establish an estimate $\bigl\Vert\int_0^\infty b(t) T_t\, dt\bigr\Vert\lesssim \Vert L_b\Vert_{{\mathcal A}_{0,S^1}({\mathbb R})}$, where $L_b$ denotes the Laplace transform of $b$. This improves previous functional calculus estimates recently obtained by the first two authors.

math.FA↗

arXiv2025

Hankel operators on $L^p(\mathbb{R}_+)$ and their $p$-completely bounded multipliers

We show that for any $1 0$. We deduce that $Hank_p(\mathbb{R}_+)$ is the dual space of$A_p(\mathbb{R}_+)$, a half-line analogue of the Figa-Talamenca-Herz algebra $A_p(\mathbb{R})$. Then we show that a function $m\colon \mathbb{R}_+^*\to \mathbb{C}$ is the symbol of a $p$-completely bounded multiplier $Hank_p(\mathbb{R}_+)\to Hank_p(\mathbb{R}_+)$ if and only if there exist $α\in L^\infty(\mathbb{R}_+;L^p(Ω))$ and $β\in L^\infty(\mathbb{R}_+;L^{p'}(Ω))$ such that $m(s+t)=\langleα(s),β(t)\rangle$ for a.e. $(s,t)\in\mathbb{R}_+^{*2}$. We also give analogues of these results in the (easier) discrete case.

math.FA↗

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

On the extension of positive maps to Haagerup non-commutative $L^p$-spaces

Let $M$ be a von Neumann algebra, let $φ$ be a normal faithful state on $M$ and let $L^p(M,φ)$ be the associated Haagerup non-commutative $L^p$-spaces, for $1\leq p\leq\infty$. Let $D\in L^1(M,φ)$ be the density of $φ$. Given a positive map $T\colon M\to M$ such that $φ\circ T\leq C_1φ$ for some $C_1\geq 0$, we study the boundedness of the $L^p$-extension $T_{p,θ}\colon D^{\frac{1-θ}{p}} M D^{\fracθ{p}}\to L^p(M,φ)$ which maps $D^{\frac{1-θ}{p}} x D^{\fracθ{p}}$ to $D^{\frac{1-θ}{p}} T(x) D^{\fracθ{p}}$ for all $x\in M$. Haagerup-Junge-Xu showed that $T_{p,\frac12}$ is always bounded and left open the question whether $T_{p,θ}$ is bounded for $θ\not=\frac12$. We show that for any $1\leq p<2$ and any $θ\in [0,2^{-1}(1-\sqrt{p-1})]\cup[2^{-1}(1+\sqrt{p-1}), 1]$, there exists a completely positive $T$ such that $T_{p,θ}$ is unbounded. We also show that if $T$ is $2$-positive, then $T_{p,θ}$ is bounded provided that $p\geq 2$ or $1\leq p<2$ and $θ\in[1-p/2,p/2]$.

math.OA↗

arXiv2025

Commuting families of polygonal type operators on Hilbert space

Let $T\colon H\to H$ be a bounded operator on Hilbert space. We say that $T$ has a polygonal type if there exists an open convex polygon $Δ\subset {\mathbb D}$, with $\overlineΔ\cap{\mathbb T}\neq\emptyset$, such that the spectrum $σ(T)$ is included in $\overlineΔ$ and the resolvent $R(z,T)$ satisfies an estimate $\Vert R(z,T)\Vert \lesssim \max\{\vert z-ξ\vert^{-1}\, :\, ξ\in \overlineΔ\cap{\mathbb T}\}$ for $z\in\overline{\mathbb D}^c$. The class of polygonal type operators (which goes back to De Laubenfels and Franks-McIntosh) contains the class of Ritt operators. Let $T_1,\ldots,T_d$ be commuting operators on $H$, with $d\geq 3$. We prove functional calculus properties of the $d$-tuple $(T_1,\ldots,T_d)$ under various assumptions involving poygonal type. The main ones are the following. (1) If the $T_k$ are contractions for all $k=1,\ldots,d$ and if $T_1,\ldots,T_{d-2}$ have a polygonal type, then $(T_1,\ldots,T_d)$ satisfies a generalized von Neumann inequality $\Vert ϕ(T_1,\ldots,T_d)\Vert \leq C\Vertϕ\Vert_{\infty,{\mathbb D}^d}$ for polynomials $ϕ$ in $d$ variables; (2) If $T_k$ is polynomially bounded with a polygonal type for all $k=1,\ldots,d$, then there exists an invertible operator $S\colon H\to H$ such that $\Vert S^{-1}T_kS\Vert \leq 1$ for all $k=1,\ldots,d$.

math.FA↗

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↗

arXiv2024

A new approach to $γ$-bounded representations

Let $X$ be a Banach space, let $(Ω,μ)$ be a $σ$-finite measure space and let $A,B\colonΩ\to B(X)$ be strongly measurable $γ$-bounded functions. We show that for all $x\in X$ and all $x^*\in X^*$, there exist a Hilbert space $K$ and two measurable functions $a_1\in L^\infty(Ω;K)$ and $a_2\in L^\infty(Ω;K)$ such that $\langle B(t)A(s)x,x^*\rangle = (a_2(t)\,\vert\, a_1(s))_{K}$ for a.e. $(s,t)$ in $Ω^2$, with $\Vert a_1\Vert_\infty \Vert a_2\Vert_\infty\leq γ(A)γ(B)\Vert x\vert\vert x^*\Vert$. This factorization property allows us to improve or simplify some results concerning $γ$-bounded representations of groups or semigroups.

math.FA↗

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↗

arXiv2021

On factorization of separating maps on noncommutative $L^p$-spaces

For any semifinite von Neumann algebra ${\mathcal M}$ and any $1\leq p<\infty$, we introduce a natutal $S^1$-valued noncommutative $L^p$-space $L^p({\mathcal M};S^1)$. We say that a bounded map $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ is $S^1$-bounded (resp. $S^1$-contractive) if $T\otimes I_{S^1}$ extends to a bounded (resp. contractive) map $T\overline{\otimes} I_{S^1}$ from $ L^p({\mathcal M};S^1)$ into $L^p({\mathcal N};S^1)$. We show that any completely positive map is $S^1$-bounded, with $\Vert T\overline{\otimes} I_{S^1}\Vert =\Vert T\Vert$. We use the above as a tool to investigate the separating maps $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ which admit a direct Yeadon type factorization, that is, maps for which there exist a $w^*$-continuous $*$-homomorphism $J\colon{\mathcal M}\to{\mathcal N}$, a partial isometry $w\in{\mathcal N}$ and a positive operator $B$ affiliated with ${\mathcal N}$ such that $w^*w=J(1)=s(B)$, $B$ commutes with the range of $J$, and $T(x)=wBJ(x)$ for any $x\in {\mathcal M}\cap L^p({\mathcal M})$. Given a separating isometry $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$, we show that $T$ is $S^1$-contractive if and only if it admits a direct Yeadon type factorization. We further show that if $p\not=2$, the above holds true if and only if $T$ is completely contractive.

math.OA↗

arXiv2021

New properties of the multivariable $H^\infty$ functional calculus of sectorial operators

This paper is devoted to the multivariable $H^\infty$ functional calculus associated with a finite commuting family of sectorial operators on Banach space. First we prove that if $(A_1,\ldots, A_d)$ is such a family, if $A_k$ is $R$-sectorial of $R$-type $ω_k\in(0,π)$, $k=1,\ldots,d$, and if $(A_1,\ldots, A_d)$ admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for some $θ_k\in (ω_k,π)$, then it admits a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for all $θ_k\in (ω_k,π)$, $k=1,\ldots,d$. Second we introduce square functions adapted to the multivariable case and extend to this setting some of the well-known one-variable results relating the boundedness of $H^\infty$ functional calculus to square function estimates. Third, on $K$-convex reflexive spaces, we establish sharp dilation properties for $d$-tuples $(A_1,\ldots, A_d)$ which admit a bounded $H^\infty(Σ_{θ_1}\times \cdots\timesΣ_{θ_d})$ joint functional calculus for some $θ_k<\fracπ{2}$.

math.FA↗

arXiv2021

Surjective separating maps on noncommutative $L^p$-spaces

Let $1\leq p<\infty$ and let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded map between noncommutative $L^p$-spaces. If $T$ is bijective and separating (i.e., for any $x,y\in L^p({\mathcal M})$ such that $x^*y=xy^*=0$, we have $T(x)^*T(y)=T(x)T(y)^*=0$), we prove the existence of decompositions ${\mathcal M}={\mathcal M}_1\mathop{\oplus}\limits^\infty{\mathcal M}_2$, ${\mathcal N}={\mathcal N}_1 \mathop{\oplus}\limits^\infty{\mathcal N}_2$ and maps $T_1\colon L^p({\mathcal M}_1)\to L^p({\mathcal N}_1)$, $T_2\colon L^p({\mathcal M}_2)\to L^p({\mathcal N}_2)$, such that $T=T_1+T_2$, $T_1$ has a direct Yeadon type factorisation and $T_2$ has an anti-direct Yeadon type factorisation. We further show that $T^{-1}$ is separating in this case. Next we prove that for any $1\leq p<\infty$ (resp. any $1\leq p\not=2<\infty$), a surjective separating map $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ is $S^1$-bounded (resp. completely bounded) if and only if there exists a decomposition ${\mathcal M}={\mathcal M}_1 \mathop{\oplus}\limits^\infty{\mathcal M}_2$ such that $T|_{L^p({\tiny {\mathcal M}_1})}$ has a direct Yeadon type factorisation and ${\mathcal M}_2$ is subhomogeneous.

math.OA↗