The Bott Suspension and the Intrinsic Join

1975 ◽  
Vol 27 (6) ◽  
pp. 1211-1221
Author(s):  
James A. Leise
Keyword(s):  
Topological Group ◽  
Relative Topology ◽  
Image Position

If (G ; U, V) is a triad with G a group we definewhere [g, u] = gug-1u-1 is the commutator. CG(U, V) will be called the (left) center of U in G modulo V or in brief a (left) C-space. If G is a topological group it will be understood that the topology on CG(U, V) is the relative topology of G.

scholarly journals Convergence of ergodic averages for many group rotations

2015 ◽  
Vol 36 (7) ◽  
pp. 2107-2120
Author(s):  
ZOLTÁN BUCZOLICH ◽  
GABRIELLA KESZTHELYI
Keyword(s):  
Topological Group ◽  
Abelian Group ◽  
Measurable Function ◽  
Dual Group ◽  
Ergodic Averages ◽  
Abelian Topological Group ◽  
Compact Abelian Topological Group ◽  
Image Position ◽  
Rotation Set ◽  
Group Rotations

Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f:G\rightarrow \mathbb{R}$ is a measurable function. Given $(n_{k})$, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages $$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$ The $f$-rotation set is $$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$We prove that if $G$ is a compact locally connected Abelian group and $f:G\rightarrow \mathbb{R}$ is a measurable function then from $m(\unicode[STIX]{x1D6E4}_{f})>0$ it follows that $f\in L^{1}(G)$. A similar result is established for ordinary Birkhoff averages if $G=Z_{p}$, the group of $p$-adic integers. However, if the dual group, $\widehat{G}$, contains ‘infinitely many multiple torsion’ then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, $f(x+n_{k}\unicode[STIX]{x1D6FC})/k$, $k=1,\ldots ,$ for almost every $x$ for many $\unicode[STIX]{x1D6FC}$; hence, some of our theorems are stated by using instead of $\unicode[STIX]{x1D6E4}_{f}$ slightly larger sets, denoted by $\unicode[STIX]{x1D6E4}_{f,b}$.

scholarly journals On Theorems of Kawada and Wendel

1958 ◽  
Vol 11 (2) ◽  
pp. 71-77 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Borel Measure ◽  
Convolution Product ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Complex Functions ◽  
Image Position ◽  
Locally Compact Topological Group ◽  
Invariant Haar Measure

Let G be a locally compact topological group, with left-invariant Haar measure. If L1(G) is the usual class of complex functions which are integrable with respect to this measure, and μ is any bounded Borel measure on G, then the convolution-product μ⋆f, defined for any f in Li byis again in L1, and

On the supports of Gauss measures on algebraic groups

1984 ◽  
Vol 96 (3) ◽  
pp. 437-445 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Topological Semigroup ◽  
Weak Topology ◽  
Algebraic Groups ◽  
Locally Compact ◽  
Compact Topological Group ◽  
Borel Measures ◽  
Image Position ◽  
Locally Compact Topological Group

For any locally compact topological group G let M(G) denote the topological semigroup of all probability (Borel) measures on G, furnished with the weak topology and with convolution as the multiplication. A Gauss semigroup on G is a homomorphism t→ μt of the strictly positive reals (under addition) into M(G) such that(i) no μt is a point mesaure,(ii) for each neighbourhood V of 1 in G we have

scholarly journals UNIVERSAL SUBGROUPS OF POLISH GROUPS

2014 ◽  
Vol 79 (4) ◽  
pp. 1148-1183 ◽  
Author(s):  
KONSTANTINOS A. BEROS
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Topological Groups ◽  
Locally Compact ◽  
Polish Groups ◽  
Countable Power ◽  
Compact Case ◽  
Image Position ◽  
The Relationship ◽  
Compactly Generated

AbstractGiven a class${\cal C}$of subgroups of a topological groupG, we say that a subgroup$H \in {\cal C}$is auniversal${\cal C}$subgroupofGif every subgroup$K \in {\cal C}$is a continuous homomorphic preimage ofH. Such subgroups may be regarded as complete members of${\cal C}$with respect to a natural preorder on the set of subgroups ofG. We show that for any locally compact Polish groupG, the countable powerGωhas a universalKσsubgroup and a universal compactly generated subgroup. We prove a weaker version of this in the nonlocally compact case and provide an example showing that this result cannot readily be improved. Additionally, we show that many standard Banach spaces (viewed as additive topological groups) have universalKσand compactly generated subgroups. As an aside, we explore the relationship between the classes ofKσand compactly generated subgroups and give conditions under which the two coincide.

On cardinal numbers associated with locally compact Abelian groups

1965 ◽  
Vol 61 (1) ◽  
pp. 75-79 ◽  
Author(s):  
J. B. Reade
Keyword(s):  
Topological Group ◽  
Abelian Groups ◽  
Cardinal Number ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Cardinal Numbers ◽  
Compact Abelian Groups ◽  
Image Position

We are concerned in this work with the following question:Suppose that i is a continuous algebraic isomorphism from the topological group H onto a subgroup of the topological group G and suppose that the image i(H) is not closed in G; then what can we say about the cardinal numberWe observe two easy results.

On nth roots and infinitely divisible elements in a connected Lie group

1981 ◽  
Vol 89 (2) ◽  
pp. 293-299 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Finite Number ◽  
Lie Groups ◽  
Lie Group ◽  
Connected Components ◽  
Compact Lie Group ◽  
Infinitely Divisible ◽  
Closed Set ◽  
Image Position ◽  
Connected Lie Group

For any group G, x ∈ G and n ∈ ℕ (the natural numbers), leti.e. the set of all nth roots of x in G. If G is a Hausdorff topological group, then Rn(x, G) is a closed set in G, but may otherwise be quite complicated. However, as we have observed in (4), if G is a compact Lie group, then Rn(x, G) always has a finite number of connected components, and this result has led us to wonder about the connectedness properties of Rn(x, G) for other Lie groups G. Here is the result.

On critical pairs of product sets in a certain matrix group

1970 ◽  
Vol 67 (3) ◽  
pp. 569-581
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Finite Measure ◽  
Matrix Group ◽  
Real Numbers ◽  
Locally Compact ◽  
Image Position ◽  
Set Function ◽  
Product Sets ◽  
Critical Pairs

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) byand, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

On the continuity of the P.S.-function

1970 ◽  
Vol 68 (2) ◽  
pp. 359-361
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Finite Measure ◽  
Locally Compact ◽  
Image Position ◽  
Set Function

Let G be a locally compact, unimodular, topological group with μ Haar measure on G, and μ* the corresponding inner measure. If (G) denotes the Borel subsets of G of finite measure, and V(G) = {μ(E):E∈(G)}, then the Product Set function, or P.S.-function, of the group G, written ΦG, is defined by

Generalized translations associated with an unbounded self-adjoint operator

1986 ◽  
Vol 99 (3) ◽  
pp. 519-528 ◽  
Author(s):  
B. Fishel
Keyword(s):  
Topological Group ◽  
Differential Operator ◽  
Normed Space ◽  
Adjoint Operator ◽  
Locally Compact ◽  
Generalized Translation ◽  
Abstract Setting ◽  
Translation Operators ◽  
Image Position ◽  
Locally Compact Topological Group

Delsarte [2], Povzner [9], Levitan [8], Leblanc [7], Dunford and Schwartz [3] (p. 1626) and Hutson and Pym [5] have discussed generalized translation operators (GTO) ‘associating with a differential operator’. The latter authors have also considered the topic in an abstract setting-the GTO ‘associates’ with a compact operator in a normed space. GTO are to have properties generalizing those of the translation operators defined by members of a group on a vector space E of functions defined on the group:(πεE, s and t are group elements). In the case of a locally compact topological group the integration spaces E = L1,L2,L∞, for a Haar measure of the group, are of especial interest.

On the Brunn-Minkowski coefficient of a locally compact unimodular group

1969 ◽  
Vol 65 (1) ◽  
pp. 33-45 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Haar Measure ◽  
Outer Measure ◽  
Real Numbers ◽  
Locally Compact ◽  
Unimodular Group ◽  
Compact Topological Group ◽  
Image Position ◽  
Locally Compact Topological Group

Let G be a locally compact topological group, and let μ be the left Haar measure on G, with μ the corresponding outer measure. If R' denotes the non-negative extended real numbers, B (G) the Borel subsets of G, and V = {μ(C):C ∈ B(G)}, then we can define ΦG: V × V → R' bywhere AB denotes the product set of A and B in G. Then clearlyso that a knowledge of ΦG will give us some idea of how the outer measure of the product set AB compares with the measures of the sets A and B.

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