All Questions
24
questions
2
votes
1
answer
259
views
Qualitative difference between "continuous" and "discontinuous" states on $M(G)$
Let $G$ be a locally compact Abelian group (we can think that $G={\mathbb R}$). Let $C_0(G)$ be the space of continuous functions $u:G\to{\mathbb C}$ vanishing at infinity with the usual $\sup$-norm, ...
- 7,366
2
votes
0
answers
79
views
An square root of the multiplicative operator on $\ell^1(\mathbb{Z}_n)$
Let us consider the finite group algebra $\ell^1(\mathbb{Z}_n)$. Let $x=(x_0,\cdots,x_{n-1})$ in $\ell^1(\mathbb{Z}_n)$ and define
$$M_x: \ell^1(\mathbb{Z}_n)\to \ell^1(\mathbb{Z}_n) : M_x(a)=a*x$$
...
- 4,030
2
votes
0
answers
149
views
A closed ideal in $L^1(T)$
Let $\mathbb{T}$ be the unit circle and consider the convolution group algebra $L^1(\mathbb{T})$. Let $I_n$ be the closed ideal generated by the polynomial $p_n(z)=z^n-1$ in $L^1(\mathbb{T})$.
Let $I=...
- 4,030
0
votes
1
answer
177
views
Irreducible sub-modules of $\ell^2(\mathbb{Z})$
It is known that $\ell^2(\mathbb{Z})$ is $\ell^1(\mathbb{Z})$-module (the module operation is the convolution).
What about the irreducible submodules? Can we characterize them?
- 4,030
4
votes
2
answers
174
views
Measure algebra on the Bohr compactification vs the bidual algebras
The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.
Let $G$ be a locally compact Abelian group and let $bG$ ...
- 11.3k
3
votes
1
answer
204
views
Relaxed/Truncated Version of Wiener's Tauberian Theorem
Background
Let $(U_t)_{t \in \mathbb{R}}$ be the (translation) $C_0$-group on $L^1(\mathbb{R})$ defined by
$$
U_t(f)(x) = f(x-t) \quad \text{for almost every } x \in \mathbb{R}
$$
(for $t \in \...
- 5,079
4
votes
1
answer
386
views
First and second cohomology groups of Banach algebras
Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" ZBL0256.18014, wrote that Guichardet in [14,15] obtained for a Banach algebra $A$,
one has $H^1(A,X)=H^2(A,X)=0$, ...
- 565
6
votes
0
answers
113
views
Homomorphisms from BV
Denote by $\mathsf{BV}(\mathbb T)$ the Banach space of functions on the circle with bounded variation which is a Banach algebra under the pointwise product. Is there a surjective homomorphism from $\...
4
votes
0
answers
262
views
Is the Gelfand transform strictly continuous?
Let $M$ be the Banach algebra of measures on the circle with $L_1$ naturally sitting as a closed ideal of $M$. Then $M$ carries the strict topology implemented by the family of seminorms $\|\mu\|_f = \...
- 113
10
votes
0
answers
203
views
Projective tensor squares of uniform algebras
In discussion with a colleague recently (Jan 2017),
$\newcommand{\AD}{A({\bf D})}\newcommand{\CT}{C({\bf T})}$
I was reminded that if $A(D)$ denotes the disc algebra and $\iota: \AD\to \CT$ is the ...
- 25.6k
4
votes
1
answer
140
views
Finding a special Banach algebra and a net of homomorphisms
If $A$ is a Banach agebra and $M$ is a Banach $A$-bimodule then a linear map $T:A\to M$ is called an $A$-module homomorphism if $$T(ab)=aT(b),\quad T(ab)=T(a)b,\qquad a,b\in A.$$ Also $A\hat{\otimes} ...
- 231
2
votes
1
answer
460
views
Ideals of $L^1(G)$
I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?
- 565
2
votes
1
answer
279
views
Does Fourier Algebra of locally compact group separate compact sets of the group?
Let $G$ be a locally compact group. Consider the left regular representation $\lambda$ over $L^2(G)$. Then according to Eymard, Fourier algebra of $G$, $A(G)$ is the set of all coefficients of $\...
- 185
4
votes
1
answer
272
views
Is the Fourier-Stieltjes algebra of a locally compact group semi-simple?
Let $G$ be a locally compact group. Is the Fourier-Stieltjes algebra $B(G)$ semi-simple?
- 4,030
5
votes
1
answer
418
views
application of factorization theorem
Young's inequlity tells us that $L^{1}(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R)$ with norm inequality
$$\|f\ast g\|_{L^{p}} \leq \|f\|_{L^1}\|g\|_{L^p};$$
and of course this ...
- 1,051