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Florin Radulescu

Publications and source records attributed to Florin Radulescu.

At least 19 recordsLinked to original sources
arXiv2023

A mixing property for the action of $\SL(3,\mathbb{Z})\times\SL(3,\mathbb{Z})$ on the Stone-Cech boundary of $\SL(3,\mathbb{Z})$

By analogy with the construction of the Furstenberg boundary, the Stone-{\v C}ech boundary of $\SL(3,\mathbb{Z})$ is a fibered space over products of projective matrices. The proximal behaviour on this space is exploited to show that the preimages of certain sequences have accumulation points which belong to specific regions, defined in terms of flags. We show that the $\SL(3,\mathbb{Z})\times \SL(3,\mathbb{Z})$-quasi-invariant Radon measures supported on these regions are tempered. Thus every quasi-invariant Radon boundary measure for $\SL(3,\mathbb{Z})$ is an orthogonal sum of a tempered measure and a measure having matrix coefficients belonging to a certain ideal $c'_0 ((\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$, slightly larger than $c_0 ((\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$. Hence the left-right representation of $C^*(\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$ in the Calkin algebra of $\SL(3,\mathbb{Z})$ factors through $C^*_{c'_0} (\SL(3,\mathbb{Z}) \times \SL(3,\mathbb{Z}))$ and the centralizer of every infinite subgroup of $\SL(3,\mathbb{Z})$ is amenable.

math.OA↗
arXiv2022

Twisted Akemann-Ostrand property for ${\rm PGL}_2(\mathbb Z[\frac{1}{p}])$ and Ramanujan Petersson Conjectures

We prove an extension of the Akemann - Ostrand theorem, regarding the simultaneous, left and right regular representations of the free group, modulo compact operators, to the case of the partial action of ${\rm PGL}_2(\mathbb Z[\frac{1}{p}]) \times {\rm PGL}_2(\mathbb Z[\frac{1}{p}])^{\rm op},$ on $\ell^2({\rm PSL}_2(\mathbb Z))$, in the presence of a non-trivial cocycle. We use this result and the operator algebra techniques developed previously to prove that the essential spectrum of the Hecke operators is contained in the bounds

math.OA↗
arXiv2022

Separable boundaries for non-hyperbolic groups

We exhibit examples of separable boundaries for non-hyperbolic groups. The main ingredient is the alignment property introduced by Furman in the study of rigidity properties of discrete subgroups of algebraic groups.

math.OA↗
arXiv2018

Benchmarking Top-K Keyword and Top-K Document Processing with T${}^2$K${}^2$ and T${}^2$K${}^2$D${}^2$

Top-k keyword and top-k document extraction are very popular text analysis techniques. Top-k keywords and documents are often computed on-the-fly, but they exploit weighted vocabularies that are costly to build. To compare competing weighting schemes and database implementations, benchmarking is customary. To the best of our knowledge, no benchmark currently addresses these problems. Hence, in this paper, we present T${}^2$K${}^2$, a top-k keywords and documents benchmark, and its decision support-oriented evolution T${}^2$K${}^2$D${}^2$. Both benchmarks feature a real tweet dataset and queries with various complexities and selectivities. They help evaluate weighting schemes and database implementations in terms of computing performance. To illustrate our bench-marks' relevance and genericity, we successfully ran performance tests on the TF-IDF and Okapi BM25 weighting schemes, on one hand, and on different relational (Oracle, PostgreSQL) and document-oriented (MongoDB) database implementations, on the other hand.

cs.DB↗
arXiv2015

Unitary representations of the Roe algebra of a discrete group and symmetries

Let $Γ$ be a discrete countable group. Consider the crossed product C$^\ast$-algebra $\mathfrak{R}(Γ) = C^{\ast}(Γ\rtimes l^{\infty}(Γ))$. Let $G$ be a larger discrete group, containing $Γ$ as an almost normal subgroup. Consequently $G$ acts by partial isomorphisms on $G$ and hence on $\mathfrak {R}(Γ)$. Let $\mathfrak{R}_G(Γ)$ be the crossed product $C^{\ast}$ - algebra $C^{\ast}(G \times (\mathfrak{R}(Γ))$. The C$^\ast$-algebra $\mathfrak{R}_G(Γ)$ has a natural representation into $\mathcal B(\ell ^2(Γ))$ and hence also admits a representation $Π_{\mathcal{Q}}$ into the Calkin algebra $\mathcal{Q}(\ell ^2(Γ))$. Let $G\rtimes Γ=Γ\times Γ^{\rm op} $ and assume that $Γ$ is exact. Assume that the non-trivial conjugation orbits under the action of $Γ$, having non amenable stabilizers, are separated, in a suitable chosen profinite topology, from the identity element in $Γ$. We also assume natural amenability conditions on the dynamics of the action of $Γ\times Γ^{\rm op}$ on cosets of amenable subgroups. Then $Π_{\mathcal Q}$ factorises to a representation of $C^{\ast}_{\rm red}(G \rtimes \mathfrak{R}(Γ))$. In particular the groups ${\mathop{SL}}_3(\mathbb Z)$, ${\mathop{\rm PGL}}_2(\mathbb Z[\frac{1}{p}])$ have the Akemann-Ostrand property. This implies, using the solidity property of Ozawa ([Oz]), that the group von Neumann algebras, $\mathcal L({\mathop{SL}}_3(\mathbb Z))$ and $\mathcal L({\mathop{SL}}_n(\mathbb Z))$, $n\geq 4$, are non-isomorphic.

math.GR↗
arXiv2015

A generalisation to Birkhoff - von Neumann theorem

The classic Birkhoff- von Neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices. In this paper, we study a generalisation of this theorem in the type $II_1$ setting. Namely, we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms, of the unit interval, with similar properties. We show that a weaker version of this theorem still holds.

math.FA↗
arXiv2015

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Consider a unitary representation $π$ of a discrete group $G$, which, when restricted to an almost normal subgroup $Γ\subseteq G$, is of type II. We analyze the associated unitary representation $\overlineπ^{\rm{p}}$ of $G$ on the Hilbert space of "virtual" $Γ_0$-invariant vectors, where $Γ_0$ runs over a suitable class of finite index subgroups of $Γ$. The unitary representation $\overlineπ^{\rm{p}}$ of $G$ is uniquely determined by the requirement that the Hecke operators, for all $Γ_0$, are the "block matrix coefficients" of $\overlineπ^{\rm{p}}$. If $π|_Γ$ is an integer multiple of the regular representation, there exists a subspace $L$ of the Hilbert space of the representation $π$, acting as a fundamental domain for $Γ$. In this case, the space of $Γ$-invariant vectors is identified with $L$. When $π|_Γ$ is not an integer multiple of the regular representation, (e.g. if $G=PGL(2,\mathbb Z[\frac{1}{p}])$, $Γ$ is the modular group, $π$ belongs to the discrete series of representations of $PSL(2,\mathbb R)$, and the $Γ$-invariant vectors are the cusp forms) we assume that $π$ is the restriction to a subspace $H_0$ of a larger unitary representation having a subspace $L$ as above. The operator angle between the projection $P_L$ onto $L$ (typically the characteristic function of the fundamental domain) and the projection $P_0$ onto the subspace $H_0$ (typically a Bergman projection onto a space of analytic functions), is the analogue of the space of $Γ$- invariant vectors. We prove that the character of the unitary representation $\overlineπ^{\rm{p}}$ is uniquely determined by the character of the representation $π$.

math.OA↗
arXiv2014

The Operator Algebra content of the Ramanujan-Petersson Problem

Let $G$ be a discrete countable group, and let $Γ$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $π$ of $G$ into the unitary group of the Hilbert space $l^2(Γ)$ that extend the left regular representation of $Γ$. Representations with this property are obtained by restricting to $G$ square integrable representations of a larger semisimple Lie group $\bar G$, containing $G$ as dense subgroup and such that $Γ$ is a lattice in $\bar G$. This type of unitary representations of of $G$ appear in the study of automorphic forms. We prove that the Ramanujan-Petersson problem regarding the action of the Hecke algebra on the Hilbert space of $Γ$-invariant vectors for the unitary representation $π\otimes \barπ$ is an intrinsic problem on the outer automorphism group of the von Neumann algebra $\mathcal L(G \rtimes L^{\infty}(\mathcal G,μ))$, where $\mathcal G$ is the Schlichting completion of $G$ and $μ$ is the canonical Haar measure on $\mathcal G$.

math.OA↗
arXiv2014

On the countable, measure preserving relation induced on an homogeneous quotient, by the action of a discrete group

We consider a countable discrete group $G$ acting ergodicaly and a.e. freely, by measure-preserving transformations, on an infinite measure space $(\mathcal X,ν)$ with $σ$-finite measure $ν$. Let $Γ\subseteq G$ be an almost normal subgroup with fundamental domain $F\subseteq \mathcal X $ of finite measure. Let $\mathcal R_G$ be the countable measurable equivalence relation on $\mathcal X$ determined by the orbits of $G$. Let $\mathcal R_G| F$ be its restriction to $F$. We find an explicit presentation, by generators and relations, for the von Neumann algebra associated, by the Feldman-Moore (\cite{FM}) construction, to the relation $\mathcal R_G|_F$. The generators of the relation $\mathcal R_G|_F$ are a set of transformations of the quotient space $F\cong \mathcal X/ Γ$, in a one to one correspondence with the cosets of $Γ$ in $G$. We prove that the composition formula for these transformations is an averaged version, with coefficients in $L^\infty(F,ν)$, of the Hecke algebra product formula (\cite{BC}). In the situation $G = PGL_2(\mathbb Z[\frac1p])$, $Γ=PSL_2(\mathbb Z)$, $p\geq 3$ prime number, the relation $\mathcal R_G|_F$ is the equivalence relation associated to a free, measure-preserving action of a free group on $(p+1)/2$ generators on $F$ (\cite{Ad},\cite {Hj}). We use the coset representations of the transformations generating $\mathcal R_G|_F$ to find a canonical treeing (\cite{Ga}).

math.OA↗
arXiv2014

On unbounded, non-trivial Hochschild cohomology in finite von Neumann algebras and higher order Berezin's quantization

We introduce a class of densely defined, unbounded, 2-Hochschild cocycles ([PT]) on finite von Neumann algebras $M$. Our cocycles admit a coboundary, determined by an unbounded operator on the standard Hilbert space associated to the von Neumann algebra $M$. For the cocycles associated to the $Γ$-equivariant deformation ([Ra]) of the upper halfplane $(Γ=PSL_2(\mathbb Z))$, the "imaginary" part of the coboundary operator is a cohomological obstruction - in the sense that it can not be removed by a "large class" of closable derivations, with non-trivial real part, that have a joint core domain, with the given coboundary.

math.OA↗
arXiv2014

Conditional expectations, traces, angles between spaces and Representations of the Hecke algebras

In this paper we extend the results in [Ra] on the representation of the Hecke algebra, determined by the matrix coefficients of a projective, unitary representation, in the discrete series of representations of the ambient group, to a more general, vector valued case. This method is used to analyze the traces of the Hecke operators. We construct representations of the Hecke algebra of a group $G$, relative to an almost normal subgroup $Γ$, into the von Neumann algebra of the group $G$, tensor matrices. The representations we obtain are a lifting of the Hecke operators to this larger algebra. By summing up the coefficients of the terms in the representation one obtains the classical Hecke operators. These representations were used in the scalar case in [Ra], to find an alternative representation of the Hecke operators on Maass forms, and hence to reformulate the Ramanujan Petersson conjectures as a problem on the angle (see e.g. A. Connes's paper [Co] on the generalization of CKM matrix) between two subalgebras of the von Neumann algebra of the group $G$: the image of the representation of the Hecke algebra and the algebra of the almost normal subgroup.

math.OA↗
arXiv2013

Cyclic Hilbert spaces and Connes' embedding problem

Let $M$ be a $II_1$-factor with trace $τ$, the linear subspaces of $L^2(M,τ)$ are not just common Hilbert spaces, but they have additional structure. We introduce the notion of a cyclic linear space by taking those properties as axioms. In Sec.2 we formulate the following problem: "does every cyclic Hilbert space embed into $L^2(M,τ)$, for some $M$?". An affirmative answer would imply the existence of an algorithm to check Connes' embedding Conjecture. In Sec.3 we make a first step towards the answer of the previous question.

math.OA↗
arXiv2013

Type II$_1$ von Neumann algebra representations of Hecke operators on Maass forms and the Ramanujan-Petersson conjectures

Classical Hecke operators on Maass forms are unitarely equivalent, up to a commuting phase, to completely positive maps on II$_1$ factors, associated to a pair of isomorphic subfactors, and an intertwining unitary. This representation is obtained through a quantized representation of the Hecke operators. in this representation, the Hecke operators act on the Berezin's quantization, deformation algebra of the fundamental domain of $\PSL(2,\Z)$ in the upper halfplane. The Hecke operators are inheriting from the ambient, non-commutative algebra on which they act, a rich structure of matrix inequalities. Using this construction we obtain that, for every prime $p$, the essential spectrum of the classical Hecke operator $T_p$ is contained in the interval $[-2\sqrt p, 2\sqrt p]$, predicted by the Ramanujan Petersson conjectures. In particular, given an open interval containing $[-2\sqrt p, 2\sqrt p]$, there are at most a finite number of possible exceptional eigenvalues lying outside this interval. The main tool for obtaining this representation of the Hecke operators (unitarely equivalent to the classical representation, up to commuting phase) is a Schurr type, positive "square root" of the state on $\PGL(2,\Q)$, measuring the displacement of fundamental domain by translations in $\PGL(2,\Q)$. The "square root" is obtained from the matrix coefficients of the discrete series representations of $\PSL(2,\R)$ restricted to $\PGL(2, \Q)$. The methods in this paper may also be applied to any finite index, modular subgroup $Γ_0(p^n)$, $n\geq 1$, of $\PSL(2,\Z)$. In this case the essential norm of the Hecke operator is equal to the norm of the corresponding convolution operator on the cosets Hilbert space $\ell^2((Γ_0(p^n))\backslash \PSL(2,\Z[1/p])$.

math.NT↗
arXiv2011

A universal, non-commutative C*-algebra associated to the Hecke algebra of double cosets

Let G be a discrete group and $Γ$ an almost normal subgroup. The operation of cosets concatanation extended by linearity gives rise to an operator system that is embeddable in a natural C* algebra. The Hecke algebra naturally embeds as a diagonal of the tensor product of this algebra with its opposite. When represented on the $l^2$ space of the group, by left and right convolution operators, this representation gives rise to abstract Hecke operators that in the modular group case, are unitarily equivalent to the classical operators on Maass wave forms

math.OA↗
arXiv2005

A non-commutative, analytic version of Hilbert's 17-th problem in type II$_1$ von Neumann algebras

We prove a non-commutative version of the Hilbert's 17th problem, giving a characterization of the class of non-commutative polynomials in n-undeterminates that have positive trace when evaluated in n-selfadjoint elements in arbitrary II1 von Neumann algebra. As a corollary we prove that Connes's embedding conjecture is equivalent to a statement that can be formulated entirely in the context of finite matrices.

math.OA↗
arXiv2004

Combinatorial aspects of Connes's embedding conjecture and asymptotic distribution of traces of products of unitaries

In this paper we study the asymptotic distribution of the moments of (non-normalized) traces $\Tr (w_1), \Tr(w_2), ..., \Tr(w_r)$, where $ w_1, w_2, >..., w_r$ are reduced words in unitaries in the group $\cU(N)$. We prove that as $N\to \infty$ these variables are distributed as normal gaussian variables $\sqrt {j_1} Z_1, ..., \sqrt{Z_r}$, where $j_1, ..., j_r$ are the number of cyclic rotations of the words $w_1, ..., w_s$ leaving them invariant. This extends a previous result by Diaconis (\cite{Diac}), where this it was proved, that $\Tr(U), \Tr(U^2), ...,$ $\Tr(U^p)$ are asymptotically distributed as $Z_1, \sqrt 2 Z_2, ..., \sqrt p Z_p$. We establish a combinatorial formula for $\int |\Tr (w_1)|^2...| \Tr(w_p)|^2$. In our computation we reprove some results from \cite{BC}.

math.OA↗
arXiv2002

The von Neumann algebra of the non-residually finite Baumslag group < a,b | a b^3 a^-1 = b^2 > embeds into R^omega

In this paper we analyze the structure of some sets of non-commutative moments of elements in a finite von Neumann algebra M. If the fundamental group of M is R_+\{0}, then the moment sets are convex, and if M is isomorphic to M tensor M, then the sets are closed under pointwise multiplication. We introduce a class of discrete groups that we call hyperlinear. These are the discrete subgroups (with infinite conjugacy classes) of the unitary group of R^omega. We prove that this class is strictly larger than the class of (i.c.c.) residually finite groups. In particular, it contains the Baumslag group < a,b | a b^3 a^-1 = b^2 >. This leads to a previously unknown (non-hyperfinite) type II_1 factor that can be embedded in R^omega. This is positive evidence for Connes's conjecture that any separable II_1 factor can be embedded into R^omega.

math.OA↗