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Vadim Alekseev

Publications and source records attributed to Vadim Alekseev.

29 records · Page 2Linked to original sources
arXiv2021

Sofic boundaries and a-T-menability

We undertake a systematic study of the approximation properties of the topological and measurable versions of the coarse boundary groupoid associated to a sequence of finite graphs of bounded degree. On the topological side, we prove that asymptotic coarse embeddability of the graph sequence into a Hilbert space is equivalent to the coarse boundary groupoid being topologically a-T-menable, thus answering a question by Rufus Willett. On the measure-theoretic side, we prove that measure-theoretic amenability resp. a-T-menability of the coarse boundary groupoid are related to hyperfiniteness and property almost-A resp. a version of "almost asymptotic embeddability into Hilbert space". These results can be directly applied to spaces of graphs coming from sofic approximations.

math.GR↗
arXiv2019

A rigidity result for normalized subfactors

We show a rigidity result for subfactors that are normalized by a representation of a lattice $Γ$ in a higher rank simple Lie group with trivial center into a finite factor. This implies that every subfactor of $LΓ$ which is normalized by the natural copy of $Γ$ is trivial or of finite index.

math.OA↗
arXiv2018

Uniqueness questions for C*-norms on group rings

We provide a large class of discrete amenable groups for which the complex group ring has several C*-completions, thus providing partial evidence towards a positive answer to a question raised by Rostislav Grigorchuk, Magdalena Musat and Mikael Rørdam.

math.OA↗
arXiv2018

On invariant random positive definite functions

We give the definition of an invariant random positive definite function on a discrete group, generalizing both the notion of an invariant random subgroup and a character. We use von Neumann algebras to show that all invariant random positive definite functions on groups with infinite conjugacy classes which integrate to the regular character are constant.

math.GR↗
arXiv2018

Sofic boundaries of groups and coarse geometry of sofic approximations

Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups. Along the way, we bring to light a coarse geometric view point on ultralimits of a sequence of finite graphs first exposed by Jan Špakula and Rufus Willett, as well as proving some bridging results concerning measure structures on topological groupoid Morita equivalences that will be of interest to groupoid specialists.

math.GR↗
arXiv2017

Cool DZ white dwarfs I: Identification and spectral analysis

White dwarfs with metal lines in their spectra act as signposts for post-main sequence planetary systems. Searching the Sloan Digital Sky Survey (SDSS) data release 12, we have identified 231 cool (<9000 K) DZ white dwarfs with strong metal absorption, extending the DZ cooling sequence to both higher metal abundances, lower temperatures, and hence longer cooler ages. Of these 231 systems, 104 are previously unknown white dwarfs. Compared with previous work, our spectral fitting uses improved model atmospheres with updated line profiles and line-lists, which we use to derive effective temperatures and abundances for up to 8 elements. We also determine spectroscopic distances to our sample, identifying two halo-members with tangential space-velocities >300 kms-1. The implications of our results on remnant planetary systems are to be discussed in a separate paper.

astro-ph.SR↗
arXiv2016

Quadratic Modules, C*-Algebras, and Free Convexity

Given a quadratic module, we construct its universal C*-algebra, and then use methods and notions from the theory of C*-algebras to study the quadratic module. We define residually finite-dimensional quadratic modules, and characterize them in various ways, in particular via a Positivstellensatz. We give unified proofs for several existing strong Positivstellensätze, and prove some new ones. Our approach also leads naturally to interesting new examples in free convexity. We show that the usual notion of a free convex hull is not able to detect residual finite-dimensionality. We thus propose a new notion of free convexity, which is coordinate-free. We characterize semialgebraicity of free convex hulls of semialgebraic sets, and show that they are not always semialgebraic, even at scalar level. This also shows that the membership problem for quadratic modules has a negative answer in the non-commutative setup.

math.OA↗
arXiv2014

On the first continuous $L^2$-cohomology of free group factors

We prove that the first continuous $L^2$-cohomology of free group factors vanishes. This answers a question by Andreas Thom regarding continuity properties of free difference quotients and shows that one can not distinguish free group factors by means of first continuous $L^2$-Betti number.

math.OA↗
arXiv2013

Measure continuous derivations on von Neumann algebras and applications to L^2-cohomology

We prove that norm continuous derivations from a von Neumann algebra into the algebra of operators affiliated with its tensor square are automatically continuous for both the strong operator topology and the measure topology. Furthermore, we prove that the first continuous L^2-Betti number scales quadratically when passing to corner algebras and derive an upper bound given by Shen's generator invariant. This, in turn, yields vanishing of the first continuous L^2-Betti number for II_1 factors with property (T), for finitely generated factors with non-trivial fundamental group and for factors with property Gamma.

math.OA↗
arXiv2012

Amenability and vanishing of L^2-Betti numbers: an operator algebraic approach

We recast the Foelner condition in an operator algebraic setting and prove that it implies a certain dimension flatness property. Furthermore, it is proven that the Foelner condition generalizes the existing notions of amenability and that the enveloping von Neumann algebra arising from a Foelner algebra is automatically injective. As an application we show how our techniques unify the previously known results concerning vanishing of L^2-Betti numbers for amenable groups, groupoids and quantum groups and moreover provides a large class of new examples of algebras with vanishing L^2-Betti numbers.

math.OA↗