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Amine Marrakchi

Publications and source records attributed to Amine Marrakchi.

At least 19 recordsLinked to original sources
arXiv2026

The classification of flows on $\mathrm{II}_1$ factors and Connes' bicentralizer problem

We settle two long-standing open problems in von Neumann algebras. First, we show that every outer flow with full Connes spectrum on the hyperfinite $\mathrm{II}_1$ factor has the Rokhlin property. By the work of Masuda and Tomatsu, such a flow is therefore unique up to cocycle conjugacy. This settles Takesaki's classification problem for flows on the hyperfinite type $\mathrm{II}_1$ factor. Drawing on type $\mathrm{III}$ theory, we develop a bicentralizer machinery for trace-preserving actions of locally compact groups. In the amenable case, we relate the bicentralizer conjecture to the Rokhlin property. For abelian groups, we prove an analog of Connes-Størmer transitivity theorem and we generalize Connes-Takesaki relative commutant theorem. A new resonance phenomenon is revealed which allows us to solve the bicentralizer conjecture for actions of $\R$. We then go back to the type $\mathrm{III}$ world and use this new resonance phenomenon to solve Connes' bicentralizer conjecture for all type $\mathrm{III}_1$ factors.

math.OA↗
arXiv2026

Kadison's problem and ergodicity of the bicentralizer flow and

We solve Kadison's problem on the existence of maximal abelian subalgebras in irreducible subfactors $N \subset M$ when $N$ is AFD or when $N$ is with expectation in $M$. In the latter case, we first show that the relative bicentralizer flow of a type $\mathrm{III}_1$ irreducible subfactor with expectation is always ergodic. This also implies that for every irreducible subfactor with expectation $N \subset M$, there exists an AFD subfactor with expectation $P \subset N$ that is irreducible in $M$.

math.OA↗
arXiv2026

Quantum steering is equivalent to state-preserving conditional expectations

In systems with infinitely many degrees of freedom, fundamental results from quantum information theory can fail. An important example is the uniqueness of purifications: Even when two subsystems, described by commuting von Neumann algebras $A$ and $B$, are tomographically complete, purifications of a state on $A$ need not be related by unitaries in $B$. It was recently shown that this occurs precisely when Haag duality fails, i.e., when the commutant $B'$ is strictly larger than $A$. This raises the question of which fundamental entanglement properties survive in such a setting. We show that, for a pure global state, the ability to steer any ensemble decomposition of the marginal state on $A$ by measurements on $B$ is equivalent to the existence of a state-preserving conditional expectation from $B'$ onto $A$. This establishes a direct connection between quantum steering and subfactor theory. The key observation is that steering is equivalent to the existence of extensions of ensemble decompositions from $A$ to $B'$. Working with general Jordan algebras, we prove that unital positive maps have state-preserving left inverses if and only if ensemble decompositions can be lifted. For the inclusion $A\hookrightarrow B'$, a left inverse is precisely a conditional expectation, yielding the characterization above.

quant-ph↗
arXiv2026

Uniqueness of almost periodic outer flows on the hyperfinite type $\mathrm{II}_1$ factor

We show that any almost periodic outer flow $α: \mathbb R \curvearrowright R$ on the hyperfinite type $\mathrm{II}_1$ factor with Connes' spectrum $Γ(α) = \mathbb R$ satisfies the Rokhlin property and thus is unique up to cocycle conjugacy. The proof relies on a key cocycle perturbation result for type $\mathrm{III}$ amenable equivalence relations. As a byproduct of our methods, we also show that every almost periodic factor of type $\mathrm{III}_1$ with separable predual has an extremal almost periodic faithful normal state.

math.OA↗
arXiv2026

Selfless W$^*$-probability spaces and Connes' bicentralizer problem

We introduce the notion of selfless W$^*$-probability space and study its connection with Connes' bicentralizer problem. In particular, we show that if $M$ is a separable type ${\rm III_1}$ factor with trivial bicentralizer, then $(M, φ)$ is selfless for every faithful normal state $φ\in M_\ast$.

math.OA↗
arXiv2025

Almost almost periodic type $\mathrm{III}_1$ factors and their 3-cohomology obstructions

We construct an exemple of a full factor $M$ such that its canonical outer modular flow $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic but $M$ has no almost periodic state. This can only happen if the discrete spectrum of $σ^M$ contains a nontrivial integral quadratic relation. We show how such a nontrivial relation can produce a 3-cohomological obstruction to the existence of an almost periodic state. To obtain our main theorem, we first strengthen a recent result of Bischoff and Karmakar by showing that for any compact connected abelian group $K$, every cohomology class in $ H^3(K,\mathbb{T})$ can be realized as an obstruction of a $K$-kernel on the hyperfinite $\mathrm{II}_1$ factor. We also prove a positive result : if for a full factor $M$ the outer modular flow $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic, then $M \otimes R$ has an almost periodic state, where $R$ is the hyperfinite $\mathrm{II}_1$ factor. Finally, we prove a positive result for crossed product factors associated to strongly ergodic actions of hyperbolic groups.

math.OA↗
arXiv2024

Kadison's problem for type III subfactors and the bicentralizer conjecture

In 1967, Kadison asked "if $N$ is a subfactor of the factor $M$ for which $N' \cap M$ consists of scalars, will some maximal abelian *-subalgebra of $N$ be a maximal abelian subalgebra of $M$?". Generalizing a theorem of Popa in the type $\mathrm{II}$ case (1981), we solve Kadison's problem for all subfactors with expectation $N \subset M$ where $N$ is either a type $\mathrm{III}_λ$ factor with $0 \leq λ< 1$ or a type $\mathrm{III}_1$ factor that satisfies Connes's bicentralizer conjecture. Our solution is based on a new explicit formula for the bicentralizer algebras of arbitrary inclusions. This formula implies a type $\mathrm{III}$ analog of Popa's local quantization principle. We generalize Haaegrup's theorem from 1984 by connecting the relative bicentralizer conjecture to the Dixmier property. Finally, we prove this conjecture for a large class of inclusions and we prove an ergodicity theorem for the bicentralizer flow. We also give applications of our methods to $\mathrm{II}_1$ factors, including a new characterization of Ozawa's W*-Akemann-Ostrand property.

math.OA↗
arXiv2024

Ergodic states on type III$_1$ factors and ergodic actions

Since the early days of Tomita-Takesaki theory, it is known that a von Neumann algebra $M$ that admits a state $φ$ with trivial centralizer $M_φ$ must be a type III$_1$ factor, but the converse remained open. We solve this problem and prove that such ergodic states form a dense $G_δ$ set among all faithful normal states on any III$_1$ factor with separable predual. Through Connes' Radon-Nikodym cocycle theorem, this problem is related to the existence of ergodic cocycle perturbations for outer group actions, which we consider in the second part of the paper.

math.OA↗
arXiv2022

Spectral gap and strict outerness for actions of locally compact groups on full factors

We prove that an outer action of a locally compact group $G$ on a full factor $M$ is automatically strictly outer, meaning that the relative commutant of $M$ in the crossed product is trivial. If moreover the image of $G$ in the outer automorphism group $\operatorname{Out} M$ is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of $M$ in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical Bogoljubov actions on free group factors or free Araki-Woods factors, these results are new.

math.OA↗
arXiv2021

Nonsingular Gaussian actions: beyond the mixing case

Every affine isometric action $α$ of a group $G$ on a real Hilbert space gives rise to a nonsingular action $\hatα$ of $G$ on the associated Gaussian probability space. In the recent paper [AIM19], several results on the ergodicity and Krieger type of these actions were established when the underlying orthogonal representation $π$ of $G$ is mixing. We develop new methods to prove ergodicity when $π$ is only weakly mixing. We determine the type of $\hatα$ in full generality. Using Cantor measures, we give examples of type III$_1$ ergodic Gaussian actions of $\mathbb{Z}$ whose underlying representation is non mixing, and even has a Dirichlet measure as spectral type. We also provide very general ergodicity results for Gaussian skew product actions.

math.DS↗
arXiv2021

Evanescent affine isometric actions and weak identity excluding groups

We investiguate a property of affine isometric actions on Hilbert spaces called evanescence. Evanescent actions are the extreme opposite of irreducible actions. Every affine isometric action decomposes naturally into an evanescent part and an irreducible part. We study when this decomposition is unique. We also study when an action that has almost fixed points is automatically evanescent. We relate these questions to the identity excluding property for groups. We also relate them to the finiteness of the von Neumann algebras generated by the linear part of the action.

math.OA↗
arXiv2020

Ergodic theory of affine isometric actions on Hilbert spaces

The classical Gaussian functor associates to every orthogonal representation of a locally compact group $G$ a probability measure preserving action of $G$ called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of $G$ on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type $\mathrm{III}_1$.

math.DS↗
arXiv2020

Isometric actions on Lp-spaces: dependence on the value of p

Answering a question by Chatterji--Druţu--Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0 2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher. We use this to show that for every connected semisimple Lie group $G$ and for every lattice $Γ< G$, we have $p_Γ=p_G$.

math.GR↗
arXiv2019

On the weak relative Dixmier property

We show that every inclusion of von Neumann algebras with a faithful normal conditional expectation has the weak relative Dixmier property. This answers a question of Popa \cite{Po99}. The proof uses an improvement of Ellis' lemma for compact convex semigroup.

math.OA↗
arXiv2019

Full factors and co-amenable inclusions

We show that if $M$ is a full factor and $N \subset M$ is a co-amenable subfactor with expectation, then $N$ is also full. This answers a question of Popa from 1986. We also generalize a theorem of Tomatsu by showing that if $M$ is a full factor and $σ\colon G \curvearrowright M$ is an outer action of a compact group $G$, then $σ$ is automatically minimal and $M^G$ is a full factor which has w-spectral gap in $M$. Finally, in the appendix, we give a proof of the fact that several natural notions of co-amenability for an inclusion $N\subset M$ of von Neumann algebras are equivalent, thus closing the cycle of implications given in Anantharaman-Delaroche's paper in 1995.

math.OA↗
arXiv2019

Tensor product decompositions and rigidity of full factors

We obtain several rigidity results regarding tensor product decompositions of factors. First, we show that any full factor with separable predual has at most countably many tensor product decompositions up to stable unitary conjugacy. We use this to show that the class of separable full factors with countable fundamental group is stable under tensor products. Next, we obtain new primeness and unique prime factorization results for crossed products coming from compact actions of higher rank lattices (e.g.\ $\mathrm{SL}(n,\mathbb{Z}), \: n \geq 3$) and noncommutative Bernoulli shifts with arbitrary base (not necessarily amenable). Finally, we provide examples of full factors without any prime factorization.

math.OA↗
arXiv2018

Full factors, bicentralizer flow and approximately inner automorphisms

We show that a factor $M$ is full if and only if the $C^*$-algebra generated by its left and right regular representations contains the compact operators. We prove that the bicentralizer flow of a type $\mathrm{III}_1$ factor is always ergodic. As a consequence, for any type $\mathrm{III}_1$ factor $M$ and any $λ\in ]0,1]$, there exists an irreducible AFD type $\mathrm{III}_λ$ subfactor with expectation in $M$. Moreover, any type $\mathrm{III}_1$ factor $M$ which satisfies $M \cong M \otimes R_λ$ for some $λ\in ]0,1[$ has trivial bicentralizer. Finally, we give a counter-example to the characterization of approximately inner automorphisms conjectured by Connes and we prove a weaker version of this conjecture. In particular, we obtain a new proof of Kawahigashi-Sutherland-Takesaki's result that every automorphism of the AFD type $\mathrm{III}_1$ factor is approximately inner.

math.OA↗