arXiv subjects

Jesse Peterson

Publications and source records attributed to Jesse Peterson.

At least 19 recordsLinked to original sources

arXiv2026

On ultraproduct approximations and property (T) factors

We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II$_1$ factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras $L(SL_3(\mathbb Z))$ and $L \mathbb F_2$ are not elementarily equivalent, and we show that the group von Neumann algebra $L\mathbb F_2$ is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II$_1$ factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which we can take to be group von Neumann algebras or group-measure space constructions.

math.OA↗

arXiv2023

Von Neumann equivalence and properly proximal groups

We introduce a new equivalence relation on groups, which we call von Neumann equivalence, that is coarser than both measure equivalence and $W^*$-equivalence. We introduce a general procedure for inducing actions in this setting and use this to show that many analytic properties, such as amenability, property (T), and the Haagerup property, are preserved under von Neumann equivalence. We also show that proper proximality, which was defined recently by Boutonnet, Ioana, and the second author using dynamics, is also preserved under von Neumann equivalence. In particular, proper proximality is preserved under both measure equivalence and $W^*$-equivalence, and from this we obtain examples of non-inner amenable groups that are not properly proximal.

math.OA↗

arXiv2023

Biexact von Neumann algebras

We introduce the notion of biexactness for general von Neumann algebras, naturally extending the notion from group theory. We show that biexactness implies solidity for von Neumann algebras, and that many of the examples of solid von Neumann algebras contained in the literature are, in fact, biexact. We also give examples of certain crossed products arising from Gaussian actions that are solid but not biexact, and we give examples of certain $q$-Gaussian von Neumann algebras that are strongly solid but not biexact. The techniques developed involve studying a certain weak form of nuclear embeddings, and we use this setting to give a new description of weak exactness for von Neumann algebras, which allows us to answer several open problems in the literature about weakly exact von Neumann algebras.

math.OA↗

arXiv2023

Some classes of smooth bimodules over II$_1$ factors and their associated 1-cohomology spaces

We study several classes of Banach bimodules over a II$_1$ factor $M$, endowed with topologies that make them "smooth" with respect to $L^p$-norms implemented by the trace on $M$. Letting $M\subset \B= \B(L^2M)$, and $2\leq p < \infty$, we consider: $(1)$ the space $\B(p)$, obtained as the completion of $\B$ in the norm \[ \vertiii{T}_p := \sup \{|\varphi(T)| \mid \varphi \in \B^*, \sup\{|\varphi(xYz)| \mid Y\in (\B)_1, x, z \in M\cap (L^pM)_1\} \leq 1 \}; \] $(2)$ the subspace $\K(p)\subset \B(p)$, obtained as the closure in $\B(p)$ of the space of compact operators $\K(L^2M)$; $(3)$ the space $\K_p\subset \B$ of operators that are $\vertiii{ \, \cdot \, }_p$-limits of bounded sequences of operators in $\K(L^2M)$. We prove that $\K_p$ are all equal to the {\it $\tau$-rank-completion} of $\K(L^2M)$ in $\B$, defined by \begin{align} \text{\rm q}\K_M:= \{K\in \B(L^2M) \mid & \exists K_n \in \K(L^2M), p_n\in \mathcal P(M), \nonumber \\ & \lim_n \|p_n(K-K_n)p_n\|= 0, \lim_n\tau(1-p_n)=0\}. \nonumber \end{align} We show that any separable II$_1$ factor $M$ admits non-inner derivations into $\text{\rm q}\K_M$, but that any derivation $\delta:M \rightarrow \text{\rm q}\K_M$ is a pointwise limit in $\tau$-rank-metric of inner derivations.

math.OA↗

arXiv2022

Properly Proximal von Neumann Algebras

We introduce the notion of proper proximality for finite von Neumann algebras, which naturally extends the notion of proper proximality for groups. Apart from the group von Neumann algebras of properly proximal groups, we provide a number of additional examples, including examples in the settings of free products, crossed products, and compact quantum groups. Using this notion, we answer a question of Popa by showing that the group von Neumann algebra of a nonamenable inner amenable group cannot embed into a free group factor. We also introduce a notion of proper proximality for probability measure preserving actions, which gives an invariant for the orbit equivalence relation. This gives a new approach for establishing strong ergodicity type properties, and we use this in the setting of Gaussian actions to expand on solid ergodicity results first established by Chifan and Ioana, and later generalized by Boutonnet. The techniques developed also allow us to answer a problem left open by Anantharaman-Delaroche in 1995, by showing the equivalence between the Haagerup property and the compact approximation property for II$_1$ factors.

math.OA↗

arXiv2022

Charmenability of arithmetic groups of product type

We discuss special properties of the spaces of characters and positive definite functions, as well as their associated dynamics, for arithmetic groups of product type. Axiomatizing these properties, we define the notions of charmenability and charfiniteness and study their applications to the topological dynamics, ergodic theory and unitary representation theory of the given groups. To do that, we study singularity properties of equivariant normal ucp maps between certain von Neumann algebras. We apply our discussion also to groups acting on product of trees.

math.GR↗

arXiv2022

Character rigidity for lattices and commensurators

We prove an operator algebraic superrigidity statement for homomorphisms of irreducible lattices, and also their commensurators, in certain higher-rank groups into unitary groups of finite factors. This extends the authors' previous work regarding non-free measure-preserving actions, and also answers a question of Connes for such groups.

math.OA↗

arXiv2022

Poisson boundaries of II$_1$ factors

We introduce Poisson boundaries of II$_1$ factors with respect to density operators that give the traces. The Poisson boundary is a von Neumann algebra that contains the II$_1$ factor and is a particular example of the boundary of a unital completely positive map as introduced by Izumi. Studying the inclusion of the II$_1$ factor into its boundary we develop a number of notions, such as double ergodicity and entropy, that can be seen as natural analogues of results regarding the Poisson boundaries introduced by Furstenberg. We use the techniques developed to answer a problem of Popa by showing that all finite factors satisfy the MV-property. We also extend a result of Nevo by showing that property (T) factors give rise to an entropy gap.

math.OA↗

arXiv2021

Cocycle superrigidity for profinite actions of irreducible lattices

Let $Γ$ be an irreducible lattice in a product of two locally compact groups and assume that $Γ$ is densely embedded in a profinite group $K$. We give necessary conditions which imply that the left translation action $Γ\curvearrowright K$ is "virtually" cocycle superrigid: any cocycle $w:Γ\times K\rightarrowΔ$ with values in a countable group $Δ$ is cohomologous to a cocycle which factors through the map $Γ\times K\rightarrowΓ\times K_0$, for some finite quotient group $K_0$ of $K$. As a corollary, we deduce that any ergodic profinite action of $Γ=\text{SL}_2(\mathbb Z[S^{-1}])$ is virtually cocycle superrigid and virtually W$^*$-superrigid, for any finite nonempty set of primes $S$.

math.DS↗

arXiv2018

Properly proximal groups and their von Neumann algebras

We introduce a wide class of countable groups, called properly proximal, which contains all non-amenable bi-exact groups, all non-elementary convergence groups, and all lattices in non-compact semi-simple Lie groups, but excludes all inner amenable groups. We show that crossed product II$_1$ factors arising from free ergodic probability measure preserving actions of groups in this class have at most one weakly compact Cartan subalgebra, up to unitary conjugacy. As an application, we obtain the first $W^*$-strong rigidity results for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.

math.OA↗

arXiv2017

Tremain equiangular tight frames

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.

math.FA↗

arXiv2016

Probably certifiably correct k-means clustering

Recently, Bandeira [arXiv:1509.00824] introduced a new type of algorithm (the so-called probably certifiably correct algorithm) that combines fast solvers with the optimality certificates provided by convex relaxations. In this paper, we devise such an algorithm for the problem of k-means clustering. First, we prove that Peng and Wei's semidefinite relaxation of k-means is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next, we show how to test the optimality of a proposed k-means solution using this dual certificate in quasilinear time. Finally, we analyze a version of spectral clustering from Peng and Wei that is designed to solve k-means in the case of two clusters. In particular, we show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model.

cs.IT↗

arXiv2015

Stabilizers of Ergodic Actions of Lattices and Commensurators

We prove that any ergodic measure-preserving action of an irreducible lattice in a semisimple group, with finite center and each simple factor having rank at least two, either has finite orbits or has finite stabilizers. The same dichotomy holds for many commensurators of such lattices. The above are derived from more general results on groups with the Howe-Moore property and property $(T)$. We prove similar results for commensurators in such groups and for irreducible lattices (and commensurators) in products of at least two such groups, at least one of which is totally disconnected.

math.DS↗

arXiv2015

Learning Boolean functions with concentrated spectra

This paper discusses the theory and application of learning Boolean functions that are concentrated in the Fourier domain. We first estimate the VC dimension of this function class in order to establish a small sample complexity of learning in this case. Next, we propose a computationally efficient method of empirical risk minimization, and we apply this method to the MNIST database of handwritten digits. These results demonstrate the effectiveness of our model for modern classification tasks. We conclude with a list of open problems for future investigation.

cs.LG↗

arXiv2015

On the tightness of an SDP relaxation of k-means

Recently, Awasthi et al. introduced an SDP relaxation of the $k$-means problem in $\mathbb R^m$. In this work, we consider a random model for the data points in which $k$ balls of unit radius are deterministically distributed throughout $\mathbb R^m$, and then in each ball, $n$ points are drawn according to a common rotationally invariant probability distribution. For any fixed ball configuration and probability distribution, we prove that the SDP relaxation of the $k$-means problem exactly recovers these planted clusters with probability $1-e^{-Ω(n)}$ provided the distance between any two of the ball centers is $>2+ε$, where $ε$ is an explicit function of the configuration of the ball centers, and can be arbitrarily small when $m$ is large.

cs.IT↗

arXiv2014

Character rigidity for special linear groups

In this paper we study characters on special linear groups SL_n(R), where R is either an infinite field or the localization of an order in a number field. We give several applications to the theory of measure preserving actions, operator-algebraic superrigidity, and almost homomorphisms.

math.FA↗

arXiv2013

Ergodicity of principal algebraic group actions

An \textit{algebraic} action of a discrete group $Γ$ is a homomorphism from $Γ$ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $Γ$ is determined by a module $M=\widehat{X}$ over the integer group ring $\mathbb{Z}Γ$ of $Γ$. The simplest examples of such modules are of the form $M=\mathbb{Z}Γ/\mathbb{Z}Γf$ with $f\in \mathbb{Z}Γ$; the corresponding algebraic action is the \textit{principal algebraic $Γ$-action} $α_f$ defined by $f$. In this note we prove the following extensions of results by Hayes \cite{Hayes} on ergodicity of principal algebraic actions: If $Γ$ is a countably infinite discrete group which is not virtually cyclic, and if $f\in\mathbb{Z}Γ$ satisfies that right multiplication by $f$ on $\ell ^2(Γ,\mathbb{R})$ is injective, then the principal $Γ$-action $α_f$ is ergodic (Theorem \ref{t:ergodic2}). If $Γ$ contains a finitely generated subgroup with a single end (e.g. a finitely generated amenable subgroup which is not virtually cyclic), or an infinite nonamenable subgroup with vanishing first $\ell ^2$-Betti number (e.g., an infinite property $T$ subgroup), the injectivity condition on $f$ can be replaced by the weaker hypothesis that $f$ is not a right zero-divisor in $\mathbb{Z}Γ$ (Theorem \ref{t:ergodic1}). Finally, if $Γ$ is torsion-free, not virtually cyclic, and satisfies Linnell's \textit{analytic zero-divisor conjecture}, then $α_f$ is ergodic for every $f\in \mathbb{Z}Γ$ (Remark \ref{r:analytic zero divisor}).

math.DS↗

arXiv2013

Phase Retrieval By Projections

The problem of recovering a vector from the absolute values of its inner products against a family of measurement vectors has been well studied in mathematics and engineering. A generalization of this phase retrieval problem also exists in engineering: recovering a vector from measurements consisting of norms of its orthogonal projections onto a family of subspaces. There exist semidefinite programming algorithms to solve this problem, but much remains unknown for this more general case. Can families of subspaces for which such measurements are injective be completely classified? What is the minimal number of subspaces required to have injectivity? How closely does this problem compare to the usual phase retrieval problem with families of measurement vectors? In this paper, we answer or make incremental steps toward these questions. We provide several characterizations of subspaces which yield injective measurements, and through a concrete construction, we prove the surprising result that phase retrieval can be achieved with $2M-1$ projections of arbitrary rank in $\HH_M$. Finally we present several open problems as we discuss issues unique to the phase retrieval problem with subspaces.

math.FA↗