scholarly journals Algebraicity of the zeta function associated to a matrix over a free group algebra

2014 ◽  
Vol 8 (2) ◽  
pp. 497-511 ◽  
Author(s):  
Christian Kassel ◽  
Christophe Reutenauer
Keyword(s):  
Zeta Function ◽  
Free Group ◽  
Group Algebra

Free pairs of symmetric and unitary units in normal subgroups of a division ring

2017 ◽  
Vol 16 (06) ◽  
pp. 1750108 ◽  
Author(s):  
Jairo Z. Goncalves
Keyword(s):  
Heisenberg Group ◽  
Free Group ◽  
Group Algebra ◽  
Division Ring ◽  
General Condition ◽  
Normal Subgroups ◽  
Unitary Subgroup

Let [Formula: see text] be the field of fractions of the group algebra [Formula: see text] of the Heisenberg group [Formula: see text], over the field [Formula: see text] of characteristic [Formula: see text]. We show that for some involutions of [Formula: see text] that are not induced from involutions of [Formula: see text], [Formula: see text] contains free symmetric and unitary pairs. We also give a general condition for a normal unitary subgroup of a division ring to contain a free group, and prove a generalization of Lewin’s Conjecture.

scholarly journals Group algebras of some generalised soluble groups

1972 ◽  
Vol 18 (1) ◽  
pp. 1-5 ◽  
Author(s):  
R. P. Knott
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Soluble Group ◽  
Group Algebras ◽  
Soluble Groups ◽  
Torsion Free Group ◽  
Open Question ◽  
Torsion Free

In (8) Stonehewer referred to the following open question due to Amitsur: If G is a torsion-free group and F any field, is the group algebra, FG, of G over F semi-simple? Stonehewer showed the answer was in the affirmative if G is a soluble group. In this paper we show the answer is again in the affirmative if G belongs to a class of generalised soluble groups

scholarly journals Algebraic reformulation of connes embedding problem and the free group algebra

2011 ◽  
Vol 181 (1) ◽  
pp. 305-315 ◽  
Author(s):  
Kate Juschenko ◽  
Stanislav Popovych
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Embedding Problem

scholarly journals Free Group Algebras in Division Rings with Valuation II

2019 ◽  
Vol 72 (6) ◽  
pp. 1463-1504
Author(s):  
Javier Sánchez
Keyword(s):  
Free Group ◽  
Group Algebra ◽  
Neumann Series ◽  
Group Algebras ◽  
Series Ring ◽  
Free Nilpotent Group ◽  
Enveloping Algebra ◽  
Division Rings ◽  
Ore Domain ◽  
Torsion Free

AbstractWe apply the filtered and graded methods developed in earlier works to find (noncommutative) free group algebras in division rings.If $L$ is a Lie algebra, we denote by $U(L)$ its universal enveloping algebra. P. M. Cohn constructed a division ring $\mathfrak{D}_{L}$ that contains $U(L)$. We denote by $\mathfrak{D}(L)$ the division subring of $\mathfrak{D}_{L}$ generated by $U(L)$.Let $k$ be a field of characteristic zero, and let $L$ be a nonabelian Lie $k$-algebra. If either $L$ is residually nilpotent or $U(L)$ is an Ore domain, we show that $\mathfrak{D}(L)$ contains (noncommutative) free group algebras. In those same cases, if $L$ is equipped with an involution, we are able to prove that the free group algebra in $\mathfrak{D}(L)$ can be chosen generated by symmetric elements in most cases.Let $G$ be a nonabelian residually torsion-free nilpotent group, and let $k(G)$ be the division subring of the Malcev–Neumann series ring generated by the group algebra $k[G]$. If $G$ is equipped with an involution, we show that $k(G)$ contains a (noncommutative) free group algebra generated by symmetric elements.

scholarly journals Noncommutative strong maximals and almost uniform convergence in several directions

2020 ◽  
Vol 8 ◽  
Author(s):  
José M. Conde-Alonso ◽  
Adrián M. González-Pérez ◽  
Javier Parcet
Keyword(s):  
Initial Data ◽  
Uniform Convergence ◽  
Free Group ◽  
Group Algebra ◽  
Orlicz Spaces ◽  
Classical Inequality ◽  
Almost Everywhere ◽  
Poisson Semigroup ◽  
Original Argument ◽  
Almost Uniform Convergence

Abstract Our first result is a noncommutative form of the Jessen-Marcinkiewicz-Zygmund theorem for the maximal limit of multiparametric martingales or ergodic means. It implies bilateral almost uniform convergence (a noncommutative analogue of almost everywhere convergence) with initial data in the expected Orlicz spaces. A key ingredient is the introduction of the $L_p$ -norm of the $\limsup $ of a sequence of operators as a localized version of a $\ell _\infty /c_0$ -valued $L_p$ -space. In particular, our main result gives a strong $L_1$ -estimate for the $\limsup $ —as opposed to the usual weak $L_{1,\infty }$ -estimate for the $\mathop {\mathrm {sup}}\limits $ —with interesting consequences for the free group algebra. Let $\mathcal{L} \mathbf{F} _2$ denote the free group algebra with $2$ generators, and consider the free Poisson semigroup generated by the usual length function. It is an open problem to determine the largest class inside $L_1(\mathcal{L} \mathbf{F} _2)$ for which the free Poisson semigroup converges to the initial data. Currently, the best known result is $L \log ^2 L(\mathcal{L} \mathbf{F} _2)$ . We improve this result by adding to it the operators in $L_1(\mathcal{L} \mathbf{F} _2)$ spanned by words without signs changes. Contrary to other related results in the literature, this set grows exponentially with length. The proof relies on our estimates for the noncommutative $\limsup $ together with new transference techniques. We also establish a noncommutative form of Córdoba/Feffermann/Guzmán inequality for the strong maximal: more precisely, a weak $(\Phi ,\Phi )$ inequality—as opposed to weak $(\Phi ,1)$ —for noncommutative multiparametric martingales and $\Phi (s) = s (1 + \log _+ s)^{2 + \varepsilon }$ . This logarithmic power is an $\varepsilon $ -perturbation of the expected optimal one. The proof combines a refinement of Cuculescu’s construction with a quantum probabilistic interpretation of M. de Guzmán’s original argument. The commutative form of our argument gives the simplest known proof of this classical inequality. A few interesting consequences are derived for Cuculescu’s projections.

scholarly journals What does a group algebra of a free group “know” about the group?

2018 ◽  
Vol 169 (6) ◽  
pp. 523-547 ◽  
Author(s):  
Olga Kharlampovich ◽  
Alexei Myasnikov
Keyword(s):  
Free Group ◽  
Group Algebra

scholarly journals Cardinality of product sets in torsion-free groups and applications in group algebras

2019 ◽  
Vol 19 (04) ◽  
pp. 2050079
Author(s):  
Alireza Abdollahi ◽  
Fatemeh Jafari
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Group Algebra ◽  
Finite Subset ◽  
Identity Element ◽  
Klein Bottle ◽  
Group Algebras ◽  
Torsion Free Group ◽  
Product Sets ◽  
Torsion Free

Let [Formula: see text] be a unique product group, i.e. for any two finite subsets [Formula: see text] of [Formula: see text], there exists [Formula: see text] which can be uniquely expressed as a product of an element of [Formula: see text] and an element of [Formula: see text]. We prove that if [Formula: see text] is a finite subset of [Formula: see text] containing the identity element such that [Formula: see text] is not abelian, then, for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Also, we prove that if [Formula: see text] is a finite subset containing the identity element of a torsion-free group [Formula: see text] such that [Formula: see text] and [Formula: see text] is not abelian, then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. Moreover, if [Formula: see text] is not isomorphic to the Klein bottle group, i.e. the group with the presentation [Formula: see text], then for all subsets [Formula: see text] of [Formula: see text] with [Formula: see text], [Formula: see text]. The support of an element [Formula: see text] in a group algebra [Formula: see text] ([Formula: see text] is any field), denoted by [Formula: see text], is the set [Formula: see text]. By the latter result, we prove that if [Formula: see text] for some nonzero [Formula: see text] such that [Formula: see text], then [Formula: see text]. Also, we prove that if [Formula: see text] for some [Formula: see text] such that [Formula: see text], then [Formula: see text]. These results improve a part of results in Schweitzer [J. Group Theory 16(5) (2013) 667–693] and Dykema et al. [Exp. Math. 24 (2015) 326–338] to arbitrary fields, respectively.

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