On the Deterministic and Asymptotic σ-Algebras of a Markov Operator

1989 ◽  
Vol 32 (1) ◽  
pp. 64-73 ◽  
Author(s):  
Ulrich Krengel ◽  
Michael Lin
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Convolution Operator ◽  
Markov Operator ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Finite Invariant Measure ◽  
The Relationship

AbstractLet P be a Markov operator on L∞(X, Σ, m) which does not disappear (i.e., P1A ≡ 0 => 1A ≡ 0 ) . We study the relationship between the σ-algebras(the deterministicσ-algebra), and the asymptoticσ-algebraWhen m is a σ-finite invariant measure, measurable iff p*npnf = f, and also iff Pnf has the same distribution as f . The case of a convolution operator on a locally compact group is considered.

S-Subgroups of the Real Hyperbolic Groups

1980 ◽  
Vol 32 (1) ◽  
pp. 246-256 ◽  
Author(s):  
Thomas J. O'Malley
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Lie Group ◽  
Closed Subgroup ◽  
Density Theorem ◽  
Locally Compact Group ◽  
Hyperbolic Groups ◽  
Locally Compact ◽  
The Real ◽  
Solvable Lie Group

IfHis a closed subgroup of a locally compact groupG, withG/Hhaving finiteG-invariant measure, then, as observed by Atle Selberg [8], for any neighborhoodUof the identity inGand any elementginG, there is an integern >0 such thatgnis inU·H·U.A subgroup satisfying this latter condition is said to be anS-sub group,or satisfiesproperty (S).IfGis a solvable Lie group, then the converse of Selberg's result has been proved by S. P. Wang [10]: IfHis a closedS-subgroup ofG,thenG/His compact. Property(S)has been used by A. Borel in the important “density theorem” (see Section 2 or [1]).

Unit Elements in the Double Dual of a Subalgebra of the Fourier Algebra A(G)

2009 ◽  
Vol 61 (2) ◽  
pp. 382-394
Author(s):  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Banach Algebra ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Arens Product ◽  
The Right ◽  
Topological Centers ◽  
The Relationship

Abstract. Let 𝒜 be a Banach algebra with a bounded right approximate identity and let ℬ be a closed ideal of 𝒜. We study the relationship between the right identities of the double duals ℬ** and 𝒜** under the Arens product. We show that every right identity of ℬ** can be extended to a right identity of 𝒜** in some sense. As a consequence, we answer a question of Lau and Ülger, showing that for the Fourier algebra A(G) of a locally compact group G, an element ϕ ∈ A(G)** is in A(G) if and only if A(G)ϕ ⊆ A(G) and Eϕ = ϕ for all right identities E of A(G)**. We also prove some results about the topological centers of ℬ** and 𝒜**.

On various types of convergence of positive definite functions on foundation semigroups

1992 ◽  
Vol 111 (2) ◽  
pp. 325-330 ◽  
Author(s):  
M. Lashkarizadeh-Bami
Keyword(s):  
Compact Group ◽  
Pointwise Convergence ◽  
Positive Definite ◽  
Limit Problem ◽  
Locally Compact Group ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
Interesting Discussion ◽  
Central Limit Problem ◽  
The Relationship

As is known, on a locally compact group G, the mere assumption of pointwise convergence of a sequence (n) of continuous positive definite functions implies uniform convergence of (n) to on compact subsets of G. This result was first proved in 1947 by Raikov8 (and independently by Yoshizawa9). An interesting discussion of the relationship between such theorems and various Cramr-Lvy theorems of the 1920s and 1930s, concerning the Central Limit Problem of probability, is given by McKennon(7, p. 62).

scholarly journals Non-minimal tree actions and the existence of non-uniform tree lattices

2004 ◽  
Vol 70 (2) ◽  
pp. 257-266
Author(s):  
Lisa Carbone
Keyword(s):  
Invariant Measure ◽  
Compact Group ◽  
Connected Graph ◽  
Locally Compact Group ◽  
Minimal Tree ◽  
Locally Compact ◽  
Discrete Subgroups ◽  
Finite Tree ◽  
Locally Finite ◽  
Uniform Tree

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 ⊆ X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ/G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.

Groups with Equal Uniformities

1969 ◽  
Vol 21 ◽  
pp. 655-659 ◽  
Author(s):  
R. T. Ramsay
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Vector Group ◽  
Locally Compact ◽  
Left And Right ◽  
The Relationship

If G = (G, τ) is a topological group with topology τ, then there is a smallest topology τ* ⊇ τ such that G* = (G, τ*) is a topological group with equal left and right uniformities (1). Bagley and Wu introduced this topology in (1), and studied the relationship between Gand G*. In this paper we prove some additional results concerning G* and groups with equal uniformities in general. The structure of locally compact groups with equal uniformities has been studied extensively. If G is a locally compact connected group, then G has equal uniformities if and only if G ≅ V× K,where F is a vector group and Kis a compact group (5). More generally, every locally compact group with equal uniformities has an open normal subgroup of the form V× K(4).

scholarly journals Interpolation in wavelet spaces and the HRT-conjecture

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Eirik Berge
Keyword(s):  
Compact Group ◽  
Wavelet Transforms ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Space Structure ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Positive Type ◽  
Time Frequency ◽  
Hrt Conjecture

AbstractWe investigate the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })\subset L^{2}(G)$$ W g ( H π ) ⊂ L 2 ( G ) arising from square integrable representations $$\pi :G \rightarrow \mathcal {U}(\mathcal {H}_{\pi })$$ π : G → U ( H π ) of a locally compact group G. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong restrictions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time–frequency analysis, this problem turns out to be equivalent to the HRT-conjecture. Finally, we consider the problem of whether all the wavelet spaces $$\mathcal {W}_{g}(\mathcal {H}_{\pi })$$ W g ( H π ) of a locally compact group G collectively exhaust the ambient space $$L^{2}(G)$$ L 2 ( G ) . We show that the answer is affirmative for compact groups, while negative for the reduced Heisenberg group.

The Superstability of the Generalized D'alembert Functional Equation

2003 ◽  
Vol 10 (3) ◽  
pp. 503-508 ◽  
Author(s):  
Elhoucien Elqorachi ◽  
Mohamed Akkouchi
Keyword(s):  
Functional Equation ◽  
Integral Equation ◽  
Compact Group ◽  
Locally Compact Group ◽  
Complex Numbers ◽  
Locally Compact

Abstract We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

scholarly journals Completely bounded bimodule maps and spectral synthesis

2017 ◽  
Vol 28 (10) ◽  
pp. 1750067 ◽  
Author(s):  
M. Alaghmandan ◽  
I. G. Todorov ◽  
L. Turowska
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Closed Subset ◽  
Spectral Synthesis ◽  
Locally Compact Group ◽  
Product Formula ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Multiplication Algebra

We initiate the study of the completely bounded multipliers of the Haagerup tensor product [Formula: see text] of two copies of the Fourier algebra [Formula: see text] of a locally compact group [Formula: see text]. If [Formula: see text] is a closed subset of [Formula: see text] we let [Formula: see text] and show that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] then [Formula: see text] is a set of local spectral synthesis for [Formula: see text]. Conversely, we prove that if [Formula: see text] is a set of spectral synthesis for [Formula: see text] and [Formula: see text] is a Moore group then [Formula: see text] is a set of spectral synthesis for [Formula: see text]. Using the natural identification of the space of all completely bounded weak* continuous [Formula: see text]-bimodule maps with the dual of [Formula: see text], we show that, in the case [Formula: see text] is weakly amenable, such a map leaves the multiplication algebra of [Formula: see text] invariant if and only if its support is contained in the antidiagonal of [Formula: see text].

Sign in / Sign up

Export Citation Format

Share Document