arXiv subjects

Raphaël Clouâtre

Publications and source records attributed to Raphaël Clouâtre.

At least 19 recordsLinked to original sources

arXiv2026

Annihilating ideals and Agler--McCarthy spectral varieties in the bidisc

The closed unit bidisc $\overline{\mathbb{D}}^2$ is known to be a spectral set for any pair $(T_1,T_2)$ of commuting contractions. When each $T_i$ is pure and has finite defect, the pair admits a much smaller spectral set: the closure of a distinguished variety $V\subset \mathbb{D}^2$. We exhibit conditions on $(T_1,T_2)$ that guarantee that no proper closed subset of $V$ can be a spectral set. In addition, we examine the relationship between $V$ and the annihilating ideal $\text{Ann}(T_1,T_2)$ in $H^\infty(\mathbb{D}^2)$. While $V$ is typically strictly larger than the zero set of $\text{Ann}(T_1,T_2)$, we isolate a natural constrained isometric co-extension $(S_1,S_2)$ of $(T_1,T_2)$ whose Taylor spectrum is contained in $V$ and is closely linked to the so-called support of $\text{Ann}(T_1,T_2)$. We also characterize when $\text{Ann}(T_1,T_2)$ is the ideal of functions vanishing on the joint point spectrum of $(S_1^*,S_2^*)$.

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arXiv2025

Rigidity of operator systems: tight extensions and noncommutative measurable structures

Let $A$ be a unital $C^*$-algebra generated by some separable operator system $S$. More than a decade ago, Arveson conjectured that $S$ is hyperrigid in $A$ if all irreducible representations of $A$ are boundary representations for $S$. Recently, a counterexample to the conjecture was found by Bilich and Dor-On. To circumvent the difficulties hidden in this counterexample, we exploit some of Pedersen's seminal ideas on noncommutative measurable structures and establish an amended version of Arveson's conjecture. More precisely, we show that all irreducible representations of $A$ are boundary representations for $S$ precisely when all representations of $A$ admit a unique "tight" completely positive extension from $S$. In addition, we prove an equivalence between uniqueness of such tight extensions and rigidity of completely positive approximations for representations of nuclear $C^*$-algebras, thereby extending the classical principle of Korovkin--Saskin for commutative algebras of continuous functions.

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arXiv2025

Null projections and noncommutative function theory in operator algebras

We study projections in the bidual of a $C^*$-algebra $B$ that are null with respect to a subalgebra $A$, that is projections $p\in B^{**}$ satisfying $|ϕ|(p)=0$ for every $ϕ\in B^*$ annihilating $A$. In the separable case, $A$-null projections are precisely the peak projections in the bidual of $A$ at which the subalgebra $A$ interpolates the entire $C^*$-algebra $B$. These are analogues of null sets in classical function theory, on which several profound results rely. This motivates the development of a noncommutative variant, which we use to find appropriate `quantized' versions of some of these classical facts. Through a delicate generalization of a theorem of Varopoulos, we show that, roughly speaking, sufficiently regular interpolation projections are null precisely when their atomic parts are. As an application, we give alternative proofs and sharpenings of some recent peak-interpolation results of Davidson and Hartz for algebras on Hilbert function spaces, also illuminating thereby how earlier noncommutative peak-interpolation theory may be applied. In another direction, given a convex subset of the state space of $B$, we characterize when the associated Riesz projection is null. This is then applied to various important topics in noncommutative function theory, such as the F.& M. Riesz property, the existence of Lebesgue decompositions, the description of Henkin functionals, and Arveson's noncommutative Hardy spaces (maximal subdiagonal algebras).

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arXiv2025

A new obstruction to Arveson's hyperrigidity conjecture

Let $A$ be a unital $C^*$-algebra containing a closed two-sided ideal $J$ and an operator system $X$. We enlarge $X$ to an operator system $\mathcal{S}(X,J)$ in $\mathbb{M}_2(A)$, and show that in order for $\mathcal{S}(X,J)$ to be hyperrigid, each $*$-representation of $C^*(X)$ annihilating $C^*(X)\cap J$ must admit a unique contractive completely positive extension from $X$ to the larger $C^*$-algebra $C^*(X)+J$. We leverage this implicit additional rigidity constraint to construct counterexamples to Arveson's hyperrigidity conjecture. A key condition in our construction is the mutual orthogonality of the atomic projection of $C^*(X)$ and the support projection of $J$, which we interpret as a new obstruction to the conjecture. Specializing to the case where $J$ is the ideal of compact operators on a Hilbert space, we recover as a by-product of our general construction the recent counterexample of Bilich and Dor-On. On the other hand, we find that such a pathology cannot be implemented using our construction when $A$ admits only finite-dimensional irreducible $*$-representations, thereby illustrating that the obstruction only manifests itself in noncommutative settings.

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arXiv2025

$C^*$-supports and abnormalities of operator systems

Let $S$ be a concrete operator system represented on some Hilbert space $H$. A $C^*$-support of $S$ is the $C^*$-algebra generated (via the Choi--Effros product) by $S$ inside an injective operator system acting on $H$. By leveraging Hamana's theory, we show that such a $C^*$-support is unique precisely when $C^*(S)$ is contained in every copy of the injective envelope of $S$ that acts on $H$. Further, we demonstrate how the uniqueness of certain $C^*$-supports can be used to give new characterizations of the unique extension property for $*$-representations, as well as the hyperrigidity of $S$. In another direction, we utilize the collection of all $C^*$-supports of $S$ to describe the subspace generated by the so-called abnormalities of $S$, thereby complementing a result of Kakariadis.

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arXiv2024

Restrictions of pure states to subspaces of $C^*$-algebras

Through the lens of noncommutative function theory, we study restrictions of pure states to unital subspaces of $C^*$-algebras, in the spirit of the Kadison--Singer question. More precisely, given a unital subspace $M$ of a $C^*$-algebra $B$, the fundamental problem is to describe those pure states $ω$ on $B$ for which $E_ω=\{ω\}$, where $E_ω$ is the set of states on $B$ extending $ω|_M$. In other words, we aim to understand when $ω|_M$ admits a unique extension to a state on $B$. We find that the obvious necessary condition that $ω|_M$ also be pure is sufficient in some naturally occurring examples, but not in general. Guided by classical results for spaces of continuous functions, we then turn to noncommutative peaking phenomena, and to the several variations on noncommutative peak points that have previously appeared in the literature. We perform a thorough analysis of these various notions, illustrating that all of them are in fact distinct, addressing their existence and, in some cases, their relative abundance. Notably, we find that none of the pre-existing notions provide a solution to our main problem. We are thus naturally led to introduce a new type of peaking behaviour for $ω$, namely that the set $E_ω$ be what we call a "pinnacle set". Roughly speaking, our main result is that $ω|_M$ admits a unique extension to $B$ if and only if $E_ω$ is an $M$-pinnacle set.

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arXiv2023

A boundary projection for the dilation order

Motivated by Arveson's conjecture, we introduce a notion of hyperrigidity for a partial order on the state space of a $C^*$-algebra $B$. We show how this property is equivalent to the existence of a boundary: a subset of the pure states which completely encodes maximality in the given order. In the classical case where $B$ is commutative, such boundaries are known to exist when the partial order is induced by some well-behaved cone. However, the relevant order for the purposes of Arveson's conjecture is the dilation order, which is not known to fit into this framework. Our main result addresses this difficulty by showing that the dilation maximal states are stable under absolute continuity. Consequently, we obtain the existence of a boundary projection in the bidual $B^{**}$, on which all dilation maximal states must be concentrated. The topological regularity of this boundary projection is shown to lie at the heart of Arveson's conjecture. Our techniques do not require $B$ to be commutative.

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arXiv2023

Lifting Sylvester equations: singular value decay for non-normal coefficients

We aim to find conditions on two Hilbert space operators $A$ and $B$ under which the expression $AX-XB$ having low rank forces the operator $X$ itself to admit a good low rank approximation. It is known that this can be achieved when $A$ and $B$ are normal and have well-separated spectra. In this paper, we relax this normality condition, using the idea of operator dilations. The basic problem then becomes the lifting of Sylvester equations, which is reminiscent of the classical commutant lifting theorem and its variations. Our approach also allows us to show that the (factored) alternating direction implicit method for solving Sylvester equaftions $AX-XB=C$ does not require too many iterations, even without requiring $A$ to be normal.

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arXiv2023

Finite-dimensional approximations and semigroup coactions for operator algebras

The residual finite-dimensionality of a $\mathrm{C}^*$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of whether residual finite-dimensionality of an operator algebra is inherited by its maximal $\mathrm{C}^*$-cover, which we resolve in many cases of interest.

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arXiv2022

Lebesgue decompositions and the Gleason--Whitney property for operator algebras

Broadly speaking, this paper is concerned with dual spaces of operator algebras. More precisely, we investigate the existence of what we call Lebesgue projections: central projections in the bidual of an operator algebra that detect the weak-$*$ continuous part of the dual space. Associated to any such projection is a Lebesgue decomposition of the dual space. We are particularly interested in Lebesgue projections in the context of inclusions of operator algebras. We show how their presence is intimately connected with an extension property for the inclusion reminiscent of a classical theorem of Gleason and Whitney. We illustrate that this Gleason--Whitney property fails in many examples of concrete operator algebras of functions, which partly explains why compatible Lebesgue decompositions are scarce, and highlights that the classical inclusion $H^\infty\subset L^\infty$ on the circle does not display generic behaviour.

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arXiv2022

Minimal boundaries for operator algebras

We study boundaries for unital operator algebras. These are sets of irreducible $*$-representations that completely capture the spatial norm attainment for a given subalgebra. Classically, the Choquet boundary is the minimal boundary of a function algebra and it coincides with the collection of peak points. We investigate the question of minimality for the non-commutative counterpart of the Choquet boundary and show that minimality is equivalent to what we call the Bishop property. Not every operator algebra has the Bishop property, but we exhibit classes of examples that do. Throughout our analysis, we exploit various non-commutative notions of peak points for an operator algebra. When specialized to the setting of $C^*$-algebras, our techniques allow us to provide a new proof of a recent characterization of those $C^*$-algebras admitting only finite-dimensional irreducible representations.

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arXiv2022

Multipliers and operator space structure of weak product spaces

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna-Pick spaces $\mathcal H$, we characterize all multipliers of the weak product space $\mathcal H \odot \mathcal H$. In particular, we show that if $\mathcal H$ has the so-called column-row property, then the multipliers of $\mathcal H$ and of $\mathcal H \odot \mathcal H$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury-Arveson space on a finite dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal H \odot \mathcal H$, which enables the use of dilations of completely bounded maps.

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arXiv2022

An abstract approach to the Crouzeix conjecture

Let $A$ be a uniform algebra, $θ:A\to M_n(\mathbb{C})$ be a continuous homomorphism and $α:A\to A$ be an antilinear contraction such that \[ \|θ(f)+θ(α(f))^*\|\le 2\|f\| \quad(f\in A). \] We show that $\|θ\|\le 1+\sqrt{2}$, and that $1+\sqrt2$ is sharp. We conjecture that, if further $α(1)=1$, then we may conclude that $\|θ\|\le2$. This would yield a positive solution to the Crouzeix conjecture on numerical ranges. In support of our conjecture, we prove that it is true in two special cases. We also discuss a completely bounded version of our conjecture that brings into play ideas from dilation theory.

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arXiv2021

Non-commutative measure theory: Henkin and analytic functionals on $\mathrm{C}^*$-algebras

Henkin functionals on non-commutative $\mathrm{C}^*$-algebras have recently emerged as a pivotal link between operator theory and complex function theory in several variables. Our aim in this paper is characterize these functionals through a notion of absolute continuity, inspired by a seminal theorem of Cole and Range. To do this, we recast the problem as a question in non-commutative measure theory. We develop a Glicksberg--König--Seever decomposition of the dual space of a $\mathrm{C}^*$-algebra into an absolutely continuous part and a singular part, relative to a fixed convex subset of states. Leveraging this tool, we show that Henkin functionals are absolutely continuous with respect to the so-called analytic functionals if and only if a certain compatibility condition is satisfied by the ambient weak-$*$ topology. In contrast with the classical setting, the issue of stability under absolute continuity is not automatic in this non-commutative framework, and we illustrate its key role in sharpening our description of Henkin functionals. Our machinery yields new insight when specialized to the multiplier algebras of the Drury--Arveson space and of the Dirichlet space, and to Popescu's noncommutative disc algebra. As another application, we make a contribution to the theory of non-commutative peak and interpolation sets.

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arXiv2021

Analytic functionals for the non-commutative disc algebra

The main objects of study in this paper are those functionals that are analytic in the sense that they annihilate the non-commutative disc algebra. In the classical univariate case, a theorem of F. and M. Riesz implies that such functionals must be given as integration against an absolutely continuous measure on the circle. We develop generalizations of this result to the multivariate non-commutative setting, upon reinterpreting the classical result. In one direction, we show that the GNS representation naturally associated to an analytic functional on the Cuntz algebra cannot have any singular summand. Following a different interpretation, we seek weak-$*$ continuous extensions of analytic functionals on the free disc operator system. In contrast with the classical setting, such extensions do not always exist, and we identify the obstruction precisely in terms of the so-called universal structure projection. We also apply our ideas to commutative algebras of multipliers on some complete Nevanlinna--Pick function spaces.

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arXiv2020

Finite-dimensionality in the non-commutative Choquet boundary: peaking phenomena and $\mathrm{C}^*$-liminality

We explore the finite-dimensional part of the non-commutative Choquet boundary of an operator algebra. In other words, we seek finite-dimensional boundary representations. Such representations may fail to exist even when the underlying operator algebra is finite-dimensional. Nevertheless, we exhibit mechanisms that detect when a given finite-dimensional representation lies in the Choquet boundary. Broadly speaking, our approach is topological and requires identifying isolated points in the spectrum of the $\mathrm{C}^*$-envelope. This is accomplished by analyzing peaking representations and peaking projections, both of which being non-commutative versions of the classical notion of a peak point for a function algebra. We also connect this question with the residual finite-dimensionality of the $\mathrm{C}^*$-envelope and to a stronger property that we call $\mathrm{C}^*$-liminality. Recent developments in matrix convexity allow us to identify a pivotal intermediate property, whereby every matrix state is locally finite-dimensional.

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arXiv2019

Gelfand transforms and boundary representations of complete Nevanlinna--Pick quotients

The main objects under study are quotients of multiplier algebras of certain complete Nevanlinna--Pick spaces, examples of which include the Drury--Arveson space on the ball and the Dirichlet space on the disc. We are particularly interested in the non-commutative Choquet boundaries for these quotients. Arveson's notion of hyperrigidity is shown to be detectable through the essential normality of some natural multiplication operators, thus extending previously known results on the Arveson--Douglas conjecture. We also highlight how the non-commutative Choquet boundaries of these quotients are intertwined with their Gelfand transforms being completely isometric. Finally, we isolate analytic and topological conditions on the so-called supports of the underlying ideals that clarify the nature of the non-commutative Choquet boundaries.

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arXiv2019

Localizable points in the support of a multiplier ideal and spectra of constrained operators

A unitarily invariant, complete Nevanlinna--Pick kernel $K$ on the unit ball determines a class of operators on Hilbert space called $K$-contractions. We study those $K$-contractions that are constrained, in the sense that they are annihilated by an ideal of multipliers. Our overarching goal is to identify various joint spectra of these constrained $K$-contractions through the vanishing locus of their annihilators. Our methods are based around a careful analysis of a subset of the ball associated to the annihilator, which we call its support. For the functional models, we show how this support completely determines several natural joint spectra. The picture is more complicated for general $K$-contractions, as their spectra can be properly contained in the support. Nevertheless, the "localizable" portion of the support always consists of spectral points. When the support is assumed to be small in an appropriate sense, we manage to effectively detect points of localizability.

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