1
vote
0answers
22 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
2
votes
0answers
19 views

Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
0
votes
0answers
10 views

How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
2
votes
0answers
46 views

How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
0
votes
0answers
17 views

How to factorize exponential as a convolution of finite number of functions( series)?

We choose a points $x_{1},..., x_{n}$ in $\mathbb Z$ (not 0), such that $$x_{k+1}\not = x_{1}+x_{2}+...+x_{k} \ \ (k=1,2,...,n-1).$$ Let $\delta_{x}$ denote the measure of total mass $1$, ...
1
vote
0answers
37 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
1
vote
1answer
32 views

Does there exists $f\in A(\mathbb T)$ such that $||f||=r$ and $||\mathrm{e}^{if}||= \mathrm{e}^{r}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t)\, \mathrm{e}^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb ...
0
votes
1answer
23 views

How to inter change of norm and limit in the Banach algebra?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
1
vote
0answers
29 views

Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
0
votes
0answers
18 views

Does there exists $f\in L^{1}(\mathbb R)\cap FL^{1}(\mathbb R)$(=Fourier algebra) but $|f|\not \in A(\mathbb R)$?

For $f\in L^{1}(\mathbb R)$; We define the Fourier transform of $f$ as follows: $$\hat{f}(\xi)= \int_{\mathbb R}f(x) e^{-2\pi i \xi x}dx; \ \text {for} \ \xi \in \mathbb R.$$ Consider a Fourier ...
2
votes
0answers
51 views

Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
8
votes
1answer
209 views

Wiener's theorem in $\mathbb{R}^n$

Reading Stein's "Singular integrals and differentiability properties of functions" I came across the following statement (this is in the proof of Lemma 3.2, pages 133-134): We now invoke the ...
1
vote
0answers
29 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 ...
2
votes
1answer
178 views

Are nilpotent Lie groups unimodular?

The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by \begin{equation*} \int_G f(xy)dx = \Delta(y)\int_Gf(x)dx \end{equation*} where $dx$ is a left Haar measure on ...
4
votes
1answer
513 views

Reference request: Fourier and Fourier-Stieltjes algebras

I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...