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-1
votes
0answers
107 views

The shape of certain (commutative) topological space

Edit: According to the comment of Simon, I revise the question as follows: Let $A=C_{0}(\mathbb{R}^{2})$. Let $B$ be the Banach subalgebra of $A$ generated by all $f\in ...
2
votes
1answer
121 views

Tangent space of the Fourier algebra $A(G)$

Let $G$ be a real Lie group and $A(G)$ be its Fourier algebra. Let us call a linear continuous functional $f:A(G)\to{\mathbb C}$ a tangent vector of $A(G)$ in the point $a\in G$, if it satisfies the ...
2
votes
1answer
112 views

When is the Fourier algebra $A(G)$ enough close to the Fourier-Stieltjes algebra $B(G)$?

I am reading P.Eymard's paper on the Fourier algebras of locally compact groups, and I have several questions about his constructions. I asked one of them in math.stackexchange, so far without ...
1
vote
2answers
105 views

The set of matrices with same spectral radius

I am working on an optimization problem over the set of positive matrices (that is, matrices where all entries are positive numbers) that have the same spectral radius. My main problem is how to ...
2
votes
0answers
43 views

Smallest left bounded approximate identity in $L_1(G)\ast\mu$

Let $G$ be a locally compact group, and $\mu\in M(G)$ be an idempotent measure. If $(e_\alpha)$ is a standard approximate identity of $L_1(G)$, then $(e_\alpha\ast\mu)$ is a left approximate identity ...
1
vote
0answers
88 views

On existence of right units with control of their norms

Does, there exist a Banach algebra with a family of right units with norms converging to 1, but without right unit of norm 1?
1
vote
0answers
51 views

On $C_0(\Omega)$-module maps from $L_\infty(\Omega,\mu)$ to $L_q(\Omega,\nu)$

Let $\Omega$ be a locally compact space, and $\mu,\nu\in C_0(\Omega)^*$. By $H_{p,q}^{B}$ (resp. $H_{p,q}^{C}$) we denote the Banach space of continuous $B(\Omega)$-module (resp. $C_0(\Omega)$-module) ...
0
votes
3answers
207 views

Specific Reference? Noncommutative topology and C^* algebras [closed]

I stumbled across this "dictionary for noncommutative topology" http://planetmath.org/noncommutativetopology and I would be very interested in learning more on the subject, particularly I'd like to ...
0
votes
0answers
126 views

$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?

Let $\mathbb T$ be a circle group. In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, ...
3
votes
1answer
81 views

Regular commutative Banach algebras which are not closed under complex conjugate

Let $A$ be a semisimple commutative Banach algebra with the maximal ideal space $X$. Further, assume that $A$ is regular i.e. for every closed set $E\subseteq X$ and $x\in X\setminus E$, there is some ...
1
vote
1answer
68 views

Unitization via “End points compactification”

We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{*}$ algebras. For example the one point compactification corresponds to the minimal ...
3
votes
1answer
285 views

Existence of certain bounded approximate identity

In trying to follow the proof of Proposition 4.11 in M. C. White, Injective modules for uniform algebras, Proc. London Math. Soc. 73 (1996) 155--184 there is a part which seems unclear. Let $I$ be ...
3
votes
1answer
138 views

$H^s(\mathbb T)$ is a Banach algebra for $s>1/2$

I have not managed to find a reference for the following fact: $H^s(\mathbb T)$ is a Banach algebra for $s>1/2$. In particular, I need reference for the following inequality: $$ \|uv\|_{H^s} ...
5
votes
2answers
218 views

When is the group algebra $L^1(G)$ semisimple?

Let $G$ be locally compact group. Define group algebra as $$L^1(G)=\{f\colon G\to\Bbb{C}\mid\int\lvert f(x)\rvert\, dx<\infty\}$$ with convolution product. When is the group algebra $L^1(G)$ ...
1
vote
1answer
81 views

Spectrum of a Banach algebra homeomorphic to the spectrum of one of its elements

In G. B. Folland - A Course in Abstract Harmonic Analysis we can read the following " (1.15) Proposition. Let be $A$ a (complex) commutative unital Banach algebra with unit $e$, let $x_0 \in A$ and ...
10
votes
1answer
327 views

Operator norms of circulant matrices

The definition and basic properties of circulant matrices can be found here: http://en.wikipedia.org/wiki/Circulant_matrix. For complex numbers $a_1,\ldots,a_n$, I will use the notation $$ ...
0
votes
1answer
155 views

Injective element of a commutative Banach algebra

A revision: According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$. Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal ...
2
votes
0answers
126 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
5
votes
1answer
245 views

NonCommutative Baire theorem

The classical Baire theorem says that the intersection of a sequence of open dense subsets of $X$, is dense, if the space is compact Hausdorff. In the language of $C^{*}$ algebras this is ...
1
vote
0answers
66 views

tensor product of the disc algebra with itself

Let $A=\mathcal{A}(\mathbb{D})$ be the disc algebra. Is there a cross norm on $A\otimes A$, which is compatible to the Banach algebra multiplication, such that the resulting Banach algebra has a ...
0
votes
0answers
164 views

A noncommutative analogy of the tube lemma

Assume that $A$ and $B$ are two unital commutative Banach algebras. Assume that $\phi \in \mathcal{M} (A)$ is an element of the maximal Ideal space. Define $\alpha: A\hat{\otimes} B \to ...
2
votes
0answers
92 views

Tauberian theorem from generalized Gelfand transform

Wiener's theorem gives the necessary and sufficient conditions for the set of translates of a set of functions to be dense in $L^1(\mathbb{R}^n)$, which translates algebraically into a statement about ...
1
vote
2answers
262 views

Amenability of $l^\infty$ [closed]

I'm working on the amenability of some Banach algebras, and I'm wondering why $l^\infty$ is amenable ? Does any one has any idea how to start ? Thank you in advance.
2
votes
1answer
186 views

ideals in the disk algebra

Let A be the disk algebra, of continuous functions on the closed disk holomorphic on the interior, with sup-norm denoted || . || . Let x be an interior point of the disk. Does there exist a ...
4
votes
0answers
148 views

$f,g , |f|f, |g|g \in A(\mathbb R) \ \text{(Banach algebra)} \implies \left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$?

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
1answer
170 views

Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banach algebra $A(\mathbb T)$ ?

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 ...
0
votes
0answers
83 views

Fourier Analysis in Kahane and Żelazko's characterization of maximal ideals in commutative Banach algebra

I've been told that Kahane and Żelazko (in A characterization of maximal ideals in commutative Banach algebra's Studia Math 29 1968) use a "non-trivial result" from Fourier analysis in their proof of ...
5
votes
2answers
441 views

When is it $C(X)$?

Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that ...
2
votes
0answers
169 views

Centralizers and containment of $c_0$

I have this question also in MSE (see: http://math.stackexchange.com/questions/666053/centralizers-and-containment-of-c-0), but I have not got an answer there. So I thought I try my luck here. Let ...
3
votes
1answer
332 views

Example of an infinite dimensional reflexive Banach algebra

If a $C^\ast$-algebra is reflexive (as a Banach space) then it is finite dimensional. Can anyone provide (or give a reference to) a nice example of an infinite dimensional non-commutative Banach ...
0
votes
0answers
58 views

Does there exists $f\in A_{\mathbb R}(\mathbb T)$ with $||f||=r$ such that $||e^{if}||=e^{r}$?

Let $\mathbb Z$, the set of integers, be a group with respect to addition and its dual group is the $\mathbb T = \{z\in \mathbb C : |z|=1\}$ , one dimensional torus. Put, $\ell^{1}(\mathbb Z)= ...
0
votes
0answers
56 views

Factorization in Fourier Algebra and its properties

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, ($\ast$ stands for a usual convolution ) $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): ...
7
votes
1answer
248 views

characterization of commutative Banach algebras

Let $A$ be a Banach algebra with the following property: For every two nets $ x_{\alpha}$ and $y_{\alpha}$ in $A$, $x_{\alpha}y_{\alpha}$ converges if and only if $y_{\alpha}x_{\alpha}$ converges. ...
0
votes
0answers
78 views

Connected component of the identity in graded banach algebras

I search for a noncommutative idempotent less Banach algebra $A$ which is graded by a finite Abelian group $G$ such that a nontrivial homogenous element lies in the same connected component ...
5
votes
1answer
210 views

$Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$

Let $F_{2}$ be the free group with two generators. Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way. ...
1
vote
1answer
100 views

Let $f \in M^{1,1} (\mathbb R)$ (Feichtinger's algebra /Modulation Space). Can we say $Fof\in M^{1,1}(\mathbb R)$; $F$ is an entire function?

The Modulation space ( Feichtinger's algebra), $$S_{0} (\mathbb R) = M^{1, 1}(\mathbb R): = \{ f\in L^{2}(\mathbb R) : V_{g}(f) \in L^{1}(\mathbb R^{2}) \};$$ where $V_{g}f (x, w)$ is the short- ...
3
votes
2answers
198 views

Banach algebra for measures induced by Haar measures

It is classical that $L^1(G, m)$ is a Banach algebra when $G$ is a locally compact group with Haar measure $m$ by using the operation of convolution via the integral ...
0
votes
0answers
117 views

Weak amenability and quasi central bounded approximate identity

Let $\cal A, \cal B$ be a non commutative Banach algebras, and $\cal A$ be weakly amenable and has a quasi central bounded approximate identity. Let $T:\cal A\to \cal B$ be an algebra ...
0
votes
1answer
218 views

Bounded approximate identity and kernel of algebra homomorphism

Let $\cal A$ be a Banach algebra with a bounded approximate identity, when a closed two sided ideal on it has a bounded approximate identity? In particular, let $\cal B$ be a Banach algebra with a ...
9
votes
2answers
374 views

Is $\mathcal{B}^{\mathbb{Z}}(l^\infty(\mathbb{Z}))$ a commutative algebra?

Consider $l^\infty(\mathbb{Z})$ the Banach space of bounded complex valued functions on the abelian group $\mathbb{Z}$ with the supremum norm. It has a natural action by $\mathbb{Z}$ given by ...
1
vote
1answer
89 views

Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces?

Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the ...
1
vote
2answers
226 views

Banach algebra of BV functions

I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.
18
votes
1answer
476 views

Kaplansky's conjecture and Martin's axiom

Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...
11
votes
1answer
323 views

$\mathbb{Z}G$ (left) Noetherian$\Rightarrow$ $l^1(G)$ is a flat $\mathbb{Z}G$-(right) module?

Let $G$ be a countable discrete group (not necessarily abelian), and suppose the group ring $\mathbb{Z}G$ is a left-Noetherian ring, for example, when $G$ is a polycyclic-by finite group. Denote the ...
1
vote
1answer
197 views

When can we “displace” an ultrafilter limit with another limit?

Let $\cal A$ be a Banach algebra, $\cal U$ be a free ultrafilter, and $\phi$ be a character. Let ${(w_{\alpha})}_{\alpha}$ be a net in $(\cal A)_{\cal U}$, and suppose that for every $(a_i)\in (\cal ...
7
votes
1answer
448 views

Banach Algebra Counterexample

Can someone give me an example of a Banach Algebra which does not have an isometric representation in a Hilbert Space ? (if possible, can you add a proof or a reference ? ) Thank you very much ! ...
2
votes
0answers
102 views

Bergmann Shilov Boundary vs Peak Points and Strong Boundary Points

Let $\Omega\subset\mathbb{C}^n$ be a bounded domain. Consider the Banach algebra $A(\Omega):=\mathcal{C}(\overline{\Omega})\cap\mathcal{O}(\Omega).$ Let $\partial_S\Omega$ denote the Bergmann Shilov ...
3
votes
2answers
227 views

Tensoring with a CAR-algebra

Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$? The answer ...
3
votes
2answers
292 views

On hyperplanes of $L\infty$

Consider the hyperplane $H=\{f\in L^\infty: \int f = 0\}$ of $L^\infty = L^\infty[0,1]$. My question is: 1. What is the Banach-Mazur distance between $H$ and $L^\infty$? Are there "natural" ...
0
votes
0answers
85 views

Spectrum space of semidirect product of a subalgebra and an ideal of a Banach Algebra

If an algebra $A$ is a semidirect product of a subalgebra $B$ and an ideal $I$. Is characterized the character space of $A$ by character space of $B$ and character space of $I$?