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3
votes
1answer
291 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
267 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
107 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 ...
13
votes
1answer
484 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
169 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
147 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
266 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
81 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
179 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
110 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
277 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
331 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
234 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
200 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 ...
5
votes
2answers
475 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
178 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
440 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
66 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)= ...
7
votes
1answer
273 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. ...
1
vote
0answers
152 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 as ...
5
votes
1answer
230 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
138 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- ...
2
votes
2answers
241 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
1answer
297 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
403 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
98 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
254 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 $BV\cap L^\infty$.
17
votes
1answer
738 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 ...
12
votes
2answers
395 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
216 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
510 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
134 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
251 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 ...
4
votes
2answers
318 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
90 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 $I$?
7
votes
1answer
247 views

Masas in second duals of Banach algebras

Lel $B$ be a Banach algebra and give $B^{**}$ one of the Arens products in order to make it a Banach algebra. Then the canonical embedding $\kappa\colon B\to B^{\ast\ast}$ is a homomorphic embedding ...
14
votes
2answers
466 views

Spectra of elements of a Banach algebra and the role played by the Hahn-Banach Theorem.

This problem was posed on Math StackExchange some time ago, but it did not garner any solutions there. I think that it is interesting enough to be posed here on Math Overflow, so here it goes. Let $ ...
10
votes
3answers
2k views

Projections in Banach spaces

Dear All, I am absolutely lost in the following problem: Let $P_s, \: s \in [0,1],$ be a uniformly bounded family of projections (idempotents) in a Banach space $X$ such that $P_s P_t = P_{{\rm ...
1
vote
1answer
151 views

Maximal spectrum of a complex, unital and commutative Banach-algebra

Let $A$ be a complex, unital and commutative Banach-algebra. Question: Is the maximal spectrum $Max(A)$ of $A$ endowed with the topology induced by the prime spectrum $Spec(A)$ of $A$, Hausdorff? ...
2
votes
0answers
154 views

Density of adjoint operators

I am interested in operators on a non-reflexive Banach space. Let $X$ be a Banach space and let $L(X)$ be the algebra of operators acting on $X$. We may embed $L(X)$ into $L(X^{\ast\ast})$ by ...
4
votes
1answer
287 views

Is an ultrapower of a faithful Banach algebra always faithful?

Let $A$ be an infinite dimensional faithful Banach algebra and let $\mathcal U$ be a free ultrafilter. Is the ultrapower $(A)_{\mathcal U}$ faithful?
2
votes
1answer
192 views

Subalgebras of $B(E)$

Let me fix an infinite-dimensional (complex) Banach space $E$. There is a cute result of Bracic and Kuzma which says that every maximal abelian subalgebra of $\mathscr{B}(E)$, the algebra of bounded ...
9
votes
0answers
365 views

Tensorial decomposition of $B(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
1
vote
0answers
159 views

Banach Algebras and the peripheral spectrum

Here is a little theorem that I'm trying to prove. I haven't seen it in literature before, but I think the applications will be quite useful, particularly in the context of Banach algebras. Denote ...
6
votes
1answer
618 views

Maximal ideals of the rings of Baire-One Functions

A real function $f:X\rightarrow \mathbb{R}$ is called Baire-one function, if there is a sequence $(f_n)_{n=1} ^\infty$ of continuous functions $f_n:X\rightarrow \mathbb{R}$ on $X$ so that for all ...
4
votes
1answer
692 views

When is a Banach Algebra $C^\star$

I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of ...
1
vote
0answers
171 views

Injective modules over Fourier algebra

Is there any article on injective modules over Fourier Algebras? Do we have anything about injectivity of $A(G)$ as a $A(G)$-bimodule?
2
votes
2answers
329 views

Ring isomorphism $\Phi \colon C(X) \to C(Y)$ and zero dimensionality of $X$

We denote the ring of all continuous real-valued functions on $X$ by $C(X)$. The ring of all bounded continuous real valued functions on $X$ is denoted by $C_b(X)$. One of the goals of the study of ...
5
votes
2answers
327 views

Terminology: Banach spaces equipped with continuous associative product?

This is admittedly a low-interest question mathematically, and is arguably a question I could resolve if I had time over the next few days to go and look through a large number of the Banach ...
23
votes
1answer
1k views

Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra?

Here's a research problem, which I think interesting. Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which satisfies $\sup_{n ...