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Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1333 ◽  
Author(s):  
Takuro Honma ◽  
Osamu Hatori

In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.


2020 ◽  
Vol 108 (3) ◽  
pp. 332-340
Author(s):  
ROGER BUNN ◽  
DAVID GROW ◽  
MATT INSALL ◽  
PHILIP THIEM

Let $p$ be an odd prime. The unary algebra consisting of the dihedral group of order $2p$, acting on itself by left translation, is a minimal congruence lattice representation of $\mathbb{M}_{p+1}$.


2020 ◽  
Vol 53 (1) ◽  
pp. 352-359
Author(s):  
Marat V. Markin

Abstract Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a scalar type spectral operator A in a complex Banach space as well as of the collection { e t A } t ≥ 0 {\{{e}^{tA}\}}_{t\ge 0} of its exponentials, which, under a certain condition on the spectrum of the operator A, coincides with the C 0 {C}_{0} -semigroup generated by A. The spectrum of A lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group { e t A } t ∈ ℝ {\{{e}^{tA}\}}_{t\in {\mathbb{R}}} of bounded linear operators generated by A. From the general results, we infer that, in the complex Hilbert space L 2 ( ℝ ) {L}_{2}({\mathbb{R}}) , the anti-self-adjoint differentiation operator A ≔ d d x A:= \tfrac{\text{d}}{\text{d}x} with the domain D ( A ) ≔ W 2 1 ( ℝ ) D(A):= {W}_{2}^{1}({\mathbb{R}}) is non-hypercyclic and so is the left-translation strongly continuous unitary operator group generated by A.


2018 ◽  
Vol 154 (7) ◽  
pp. 1333-1361 ◽  
Author(s):  
Friedrich Martin Schneider ◽  
Andreas Thom

We extend Følner’s amenability criterion to the realm of general topological groups. Building on this, we show that a topological group $G$ is amenable if and only if its left-translation action can be approximated in a uniform manner by amenable actions on the set $G$. As applications we obtain a topological version of Whyte’s geometric solution to the von Neumann problem and give an affirmative answer to a question posed by Rosendal.


2017 ◽  
Vol 25 (2) ◽  
pp. 99-135
Author(s):  
Rory Biggs

Abstract We investigate the isometry groups of the left-invariant Riemannian and sub-Riemannian structures on simply connected three-dimensional Lie groups. More specifically, we determine the isometry group for each normalized structure and hence characterize for exactly which structures (and groups) the isotropy subgroup of the identity is contained in the group of automorphisms of the Lie group. It turns out (in both the Riemannian and sub-Riemannian cases) that for most structures any isometry is the composition of a left translation and a Lie group automorphism.


Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2721-2730
Author(s):  
Ali Ghaffari ◽  
Somayeh Amirjan

Let G be a locally compact group and let ? be a closed subgroup of G x G. In this paper, the concept of commutativity with respect to a closed subgroup of a product group, which is a generalization of multipliers under the usual sense, is introduced. As a consequence, we obtain characterization of operators on L2(G) which commute with left translation when G is amenable.


2015 ◽  
Vol 99 (3) ◽  
pp. 399-414 ◽  
Author(s):  
ALEN OSANÇLIOL ◽  
SERAP ÖZTOP

For a locally compact group $G$ with left Haar measure and a Young function ${\rm\Phi}$, we define and study the weighted Orlicz algebra $L_{w}^{{\rm\Phi}}(G)$ with respect to convolution. We show that $L_{w}^{{\rm\Phi}}(G)$ admits no bounded approximate identity under certain conditions. We prove that a closed linear subspace $I$ of the algebra $L_{w}^{{\rm\Phi}}(G)$ is an ideal in $L_{w}^{{\rm\Phi}}(G)$ if and only if $I$ is left translation invariant. For an abelian $G$, we describe the spectrum (maximal ideal space) of the weighted Orlicz algebra and show that weighted Orlicz algebras are semisimple.


Author(s):  
Maciej Dunajski

We propose a construction of Skyrme fields from holonomy of the spin connection of gravitational instantons. The procedure is implemented for Atiyah–Hitchin and Taub–NUT instantons. The skyrmion resulting from the Taub–NUT is given explicitly on the space of orbits of a left translation inside the whole isometry group. The domain of the Taub–NUT skyrmion is a trivial circle bundle over the Poincaré disc. The position of the skyrmion depends on the Taub–NUT mass parameter, and its topological charge is equal to two.


2012 ◽  
Vol 49 (3) ◽  
pp. 301-314
Author(s):  
Ali Ghaffari

Suppose that A is either the Banach algebra L1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.


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