1
vote
0answers
29 views

Family of equivalent unitary representations is not a set.

I have recently come across a statement in the book: Kazhdan's property (T) by B. Bekka, P. de la Harpe, A. Valette at the beginning Appendix F.2. Fell topology on sets of unitary representations. ...
1
vote
1answer
43 views

Open subgroup and group algebra

Let $G$ be a locally compact group and $H$ be an open subgroup of $G$. Consider the group algebras $L^1(G)$ and $L^1(H)$ with convolution product and consider $L^1(H)$ as a subalgebra of $L^1(G)$ ...
0
votes
1answer
43 views

how to deduce that finite group $G$ has at most $|G|$ characters

Let $G$ be a finite abelian group. A character of $G$ is a group homomorphism $\chi: G\longrightarrow \mathbb{C}^{\times}$. I have proved by induction that for distinct $\chi_1,\cdots,\chi_r$, they ...
5
votes
1answer
384 views

Open subgroup of $SO(3)$

Does $SO(3)$ have an open nontrivial subgroup?(Group $SO(3)$ with usual matrices product, is all $3\times 3$ matrices whose determinant is 1 and for every element $A\in SO(3)$ we have $A^tA=AA^t=I_3$ ...
1
vote
0answers
37 views

Definite positive measure and GNS representation

Let $G$ be a locally compact group. Let $\mu$ be a positive definite complex measure ([D, p295]): we have $\mu(f*f^*)\geq 0$ for any compact support continuons function $f \in C_c(G)$. In [D, p ...
1
vote
0answers
63 views

Fourier algebra and multipliers of a finite discrete group

Let $G$ be a finite discrete group. We denote by $A(G)$ the Fourier algebra of $G$ and $M_{cb}A(G)$ the space of completely bounded multipliers of $A(G)$. Is is true that $A(G)=M_{cb}A(G)$ ...
2
votes
1answer
104 views

The various central extensions of $(G\times G)$ by $T$.

Let $G$ be a locally compact abelian group, isomorphic to $G^*$, its Pontryagin Dual. Let $T$ denote the unit circle in $\mathbb{C}$, where continuous morphisms $\chi: G\to T$ are the elements of ...
5
votes
1answer
264 views

Harmonic functions on $\mathbf{Z}^2$

Problem 1: Find all functions $f:\mathbf{Z}^2 \to \mathbf{R}$ which are harmonic in the sense that $$f(x,y) = \frac{f(x+1,y) + f(x-1,y) + f(x,y+1) + f(x,y-1)}{4}$$for all $(x,y)\in\mathbf{Z}^2$, ...
0
votes
1answer
63 views

What is the “spectrum of $L^1(G)$”?

If $G$ is a locally compact abelian group, what does "the spectrum of $L^1(G)$ mean?" This comes from Folland's A Course in Abstract Harmonic Analysis. As I understand it, $L^1(G)$ is the integrable ...
3
votes
1answer
112 views

Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?

Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$. An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space. What are the necessary and ...
3
votes
1answer
87 views

Reference for: $G$ discrete iff the measure algebra $M(G)$ is weakly amenable.

I search the reference for the proof of the following theorem: Let $G$ be a locally compact group. Then the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable. The ...
3
votes
0answers
55 views

$\Lambda_p$-set for compact abelian group

We denote by $|A|$ the cardinal of a set $A$. Let $S$ be a subset of $\mathbb{Z}$. Denote $S_N=S\cap [0,N]$ where $N$ is an integer. Suppose $2<p<\infty$. There is well-known that if $S$ is a ...
0
votes
1answer
255 views

Transitive group actions and homogeneous spaces

Given a topological group $G$ and a space $X$ with a transitive $G$ action, let $G_x$ be the isotropy group of a point. In Folland "A course in harmonic analysis", there is a statement that $X$ is ...
3
votes
1answer
114 views

Fourier analysis on groups and paths in a Cayley graph

If one takes a cyclic group and a function on this group, and performs harmonic analysis on it (classical Fourier analysis), the result is a set of coefficients, each one of them corresponding to ...
2
votes
1answer
316 views

Convolution on group with measure

I was wondering about the generalization of the concept of convolution from the familiar one on real spaces and how many properties still remain. For convolution on Lebesgue-integrable real-valued ...