All Questions
Tagged with topological-groups harmonic-analysis
68 questions
2
votes
1
answer
257
views
Haar measures of compact subgroups
Let $G$ be a locally compact group, $K$ a compact subgroup in $G$, and $\mu_K$ the normalized Haar measure on $K$:
$$
\mu_K(K)=1.
$$
Let us denote by $\widetilde{\mu_K}$ the measure on $G$ defined as ...
- 7,426
2
votes
2
answers
68
views
An [IN$]_B$ group with a non-normal compact $B$-invariant subgroup
Let $G$ be a locally compact group with the group of topological group automorphisms $Aut(G)$ furnished with the compact-open topology. Let $B$ be a subgroup of $Aut(G)$. We call $G$ an [IN$]_B$ if ...
- 299
2
votes
0
answers
187
views
How can the maximal ideal space of the Fourier Stieltjes algebra be non-separable?
I have been asking a fair few (probably elementary) questions about abstract harmonic analysis lately. By means of explanation, I am just feeling around the subject at the moment and trying to build ...
- 1,955
2
votes
0
answers
435
views
Generalized conjugacy classes in (topological) groups
Let $G$ be a topological group. We define an equivalence relation on $G$ as follows:
For $a,b\in G$ we set $a\sim b$ if the following two maps are topologically conjugate:
$$x\mapsto ax,\qquad x\...
- 356
2
votes
0
answers
157
views
Primitive ideal space and unitary dual of a [SIN] group - when are they Hausdorff?
Recall that a locally compact group $G$ is said to be
an $[FC]^-$ group, if each conjugacy class in $G$ has a compact closure;
an $[SIN]$ group, if each neighborhood of the identity includes a ...
- 399
2
votes
0
answers
807
views
Why groups that admit Folner Sequences are amenable
I've been looking at Folner's Condition recently, and I'm struggling to find a proof for why the existence of a Folner sequence on a locally compact group implies that it is amenable (and the converse ...
- 21
1
vote
0
answers
154
views
measure of Haar
Let $(G,K)$ be a Gelfand pair.
Why, for a function $f$ $K$-binvariant with respect to a compact subgroup $K$ of a group $G$, do we have the following equality:
$$ f(xy) = \int_K f(xky) \, dk$$
A ...
- 109
1
vote
0
answers
112
views
Idempotent conjecture and non-abelian solenoid
Is there a discrete non-abelian group whose dual in a reasonable sense is isomorphic to the solenoid constructed via a sequence of quaternions $S^3$ instead of a sequence of circles? The motivation ...
- 356
1
vote
1
answer
647
views
Haar measure coming from Pontryagin duality v/s Fourier inversion
Not research but advertising this question from mse in case someone wants to answer.
I'm struggling with some bookkeeping associated with the Pontryagin duality theorem. I'm thinking about the first ...
- 675
1
vote
0
answers
145
views
How a profinite group can be obtained from its normal open subgroups?
Let $\Delta$ be a set, each element of which is a profinite group (2 distinct elements of $\Delta$ may be isomorphic). Under what conditions on $\Delta$, there exists a profinite group $G$ which has $\...
- 2,118
1
vote
0
answers
143
views
Irreducible unitary representations of discrete abelian groups
It seems to me that the statement below should be true but I would like to double-check.
Statement: Let $H$ be a (separable) complex Hilbert space and consider its associated unitary group $U(H)$ ...
- 7,609
1
vote
0
answers
115
views
Can Gaussian measure be characterized by unitary representations?
It is well known that Fourier transform switches positive-definite functions with positive measures on a (locally compact topological) group. Further, the positive definite functions can be ...
- 6,249
0
votes
2
answers
157
views
Is there an integer sequence $(k_n)$ where each term is non-zero such that $\lim z^{k_n}=1$ for every point z on the unit circle?
Is there an integer sequence $(k_n)$ where each term is non-zero such that $\lim z^{k_n}=1$ for every point $z$ on the unit circle of the complex plane? I don't think it exists，but I don't know how to ...
- 83
0
votes
2
answers
477
views
Basic question on minimal flows
I know that minimal flows are actions for which no proper closed invariant subsets exist, but I am unclear how to understand this concept.
If a coset flow on a quotient space Gamma/S is ergodic, ...
- 3
0
votes
0
answers
96
views
Idempotent conjecture and (weak) connectivity of (a reasonable) dual group
What is an example of a torsion free discrete abelian group $G$ whose dual space $\hat{G}$ is not a path connected space?
The Motivation: The motivation comes from the idempotent conjecture of ...
- 356
0
votes
0
answers
97
views
How large this subset is to say that it should equal the group?
Let $\alpha$ be a continuous automorphism on a compact group $G$ with normalized Haar measure $m$. We may say $\alpha$ is $n$-splitting, if the set
$$\text{Spl}_n(\alpha):=\left\{g\in G: \prod_{k=1}^...
- 2,118
0
votes
0
answers
54
views
Left introversion operators associated to function spaces on semigroups
I am stuck on the following question for quite sometime now. Please help, any comment is welcome.
Let $S$ be a topological semigroup and $\mathcal{F}$ be a translation invariant, conjugate closed ...
- 1
-2
votes
1
answer
118
views
Mismatch between equivalent definitions of the Bohr compactification of the reals
I feel I'm overlooking something very silly.
The Bohr compactification of $\mathbb R$ has two equivalent definitions.
The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under ...
- 1,955