China
Africa
Explore chapters and articles related to this topic
Group Theory
Published in Gregory S. Chirikjian, Alexander B. Kyatkin, Engineering Applications of Noncommutative Harmonic Analysis, 2021
Gregory S. Chirikjian, Alexander B. Kyatkin
DEFINITION 7.3A Hausdorff space4X is called locally compact if for each x∈ X and every open set U containing xthere exists an open set W such that Cl(W) is compact and x∈ W⊆Cl(W)⊆U. A locally compact group is a group for which the underlying set is locally compact [17, 28].
Functional Equations on Affine Groups
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
We note that neither O(p, q) nor SO(p, q) is compact if pq ≠ 0. Nevertheless, by the Cartan–Iwasawa–Malcev Theorem (see e.g. Iwasawa (1949); Malcev (1945); Stroppel (2006)), every connected Lie group (in fact, every connected locally compact group) has maximal compact subgroups which are conjugate to each other. Hence maximal compact subgroups are in some sense unique. For instance, a maximal compact subgroup in O(p, q) is isomorphic to O(p) × O(q), and similarly, a maximal compact subgroup in SO(p, q) is isomorphic to SO(p) × SO(q). Here O(p) × O(q), respectively, SO(p) × SO(q) acts on ℝp ⊕ ℝq “componentwise”: the O(p) part, respectively the SO(p) part acts on ℝp, and the O(q) part, respectively, the SO(q) part acts on ℝq. In particular, a maximal compact subgroup of the proper Lorentz group is isomorphic to SO(1)×SO(3)≅SO(3), which acts on ℝ ⊕ ℝ3 by the rule: for k in SO(3) we have k⋅(t,x,y,z)=(t,[k]⋅(xyz)),
A dynamical proof of the van der Corput inequality
Published in Dynamical Systems, 2022
Nikolai Edeko, Henrik Kreidler, Rainer Nagel
It would be interesting to also apply our approach to closed subsemigroups of a locally compact group with right Haar measure m having a topological right Følner net (e.g. and ), i.e. is compact with positive measure for every and for every (cf. [24, Examples 1.2 (e)]). In particular, this could lead to a new proof of [2, Theorem 2.12].
Measures and entropy on non compact dynamics
Published in Dynamical Systems, 2020
Let G be a locally compact group and let be a G-space, that is, G acts as a group of invertible measure-preserving transformations of X. In particular, the action lifts to a continuous representation of G on . To each subgroup , one could associate the so-called H-orbit partition , generating the algebra satisfying conditions – listed in [44, 2.2]. Let . We say that the measure-preserving transformation Tσ-commutes with the G-action, if , for . In this case, is T-invariant. Same is true for , when H is σ-invariant, i.e. . If H is a σ-invariant closed normal subgroup of G, we consider the space as a G/H-space (as H acts trivially on the union of the elements of the partition ), where is the restriction of μ to . Since is measure preserving, T induces a transformation on which -commutes with the G/H-action. If H is -invariant, then T-commutes with the H-action.