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Questions tagged [banach-algebras]

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28 views

An example of a Banach algebra with a specified property

I asked this question (https://math.stackexchange.com/questions/3076735/an-example-of-a-banach-algebra-satisfying-given-conditions) but unfortunately no one answered it. Please help me to find an ...
0
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1answer
71 views

The definiton of a multiplier on a Banach algebra

Let $A$ be a Banach algebra. Some textbooks define a (left ) multiplier as a map $T:A\rightarrow A$ satisfying $T(ab)=T(a)b$ for all $a,b\in A$ and assume that $A$ needs to be a without order Banach ...
6
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0answers
89 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 $\...
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0answers
48 views

There are many discontinuous Point derivations on the Banach algebra $(\mathbb{C^{(n)}[0,1], \|\|_n})$

This is an Exercise 6.2.55 in Garth Dales , Introduction to Banach algebra Show that there are many discontinuous Point derivations on the Banach algebra $(\mathbb{C^{(n)}[0,1], \|\|_n})$ where ...
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0answers
71 views

Automatic continuity in Banach algebras

I have the following two questions 1: Let $A$ and $B$ be Banach algebras and suppose that $B$ is semisimple. Let $T:A \to B $ be a homomorphism with $\overline {TA}=B.$ Is $T$ automatically ...
4
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0answers
124 views

Gelfand spectrum as a measure space

Given a Lebesgue probability measure space $(X,m)$ (say, just the unit interval with the Lebesgue measure on it), let $A$ be a closed subalgebra of the real $L^\infty(X,m)$. Then one can realize the ...
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0answers
50 views

Evaluate $\operatorname{Rad}(A/\operatorname{Rad}(A))$ in a Banach algebra

I've asked this question here Let $A$ be a Banach algebra with identity $e_A$, I'd like to find $\operatorname{Rad}(A/\operatorname{Rad}(A)).$ whre we define $\operatorname{Rad}(A)=\{a\in A:...
3
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0answers
53 views

Characterizing the separability of the Gelfand space of a semisimple commutative Banach algebra

Problem. Is the separability of the Gelfand space of a semi-simple commutative Banach algebra $A$ equivalent to the existence of a countable family $\{\varphi_n\}_{n\in\omega}$ of multiplicative ...
9
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2answers
233 views

Explicit proof that $c_0$-module $\ell_\infty$ is not projective

It is well known in narrow circles that the homological dimension (in the sense of relative Banach homology) of $c_0$-module $\ell_\infty$ is 2. As the corollary, this module is not projective. This ...
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0answers
64 views

Extension of a derivation

Let I be a closed left ideal of a Banach algebra A and let D:I\to I* be a derivation. Does D extend to a derivation from A to A*?
10
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1answer
588 views

Quantum functional analysis

Can one explain some philosophy behind "quantum functional analysis" (or "quantized functional analysis") which was initiated and developed by such researchers as: Ruan Z.-J., Pisier J., Effros E.G., ...
10
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1answer
264 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...
0
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1answer
65 views

Regarding $\ell_p$ direct sums

I am reading this paper by S.H Karin titled Norm attaining operators and pseudospectrum. In page 2 he gives the definition of $l_p$ direct sum of a family of Banach spaces as follows: If $1\leq p< \...
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0answers
50 views

Proof of Lemma 7.1 Bonsall and Duncan

In the proof of Lemma 1 in section 7 (A functional calculus for single Banach algebra element) of the book Complete normed algebras by Bonsall and Duncan, the last line says $$\phi\left(\frac{1}{2\pi ...
6
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0answers
63 views

Are invertible measures strictly dense?

Let $L_1(\mathbb T)$ be considered as a closed ideal of $M(\mathbb T)$, the Banach algebra of measures on the circle. Then $M(\mathbb T)$ can be identified with the multiplier algebra of $L_1(\mathbb ...
0
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1answer
124 views

Regarding exponential in a Banach algebra

Let $A$ be a complex unital Banach algebra. Let exp$(A)$ denote the range of the exponential function on $A$. Now exp$(A)$ lies in the set of all invertible elements of $A$ (denoted by $G(A)$). Can ...
5
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1answer
201 views

Model theory of Banach algebras

Let us consider the (metric) theory of Banach algebras. I have a sentence encoding the (possible) openness of multiplication in a given Banach algebra: $$(\forall x) (\forall y) (\forall \varepsilon &...
4
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0answers
159 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 = \...
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0answers
17 views

Admissible Quadruple of Type L for locally compact Hausdorff space

Hatori and Oi defined admissible quadruple of type L at definition $4,$ page $6.$ Let $X$ and $Y$ be compact Hausdorff spaces. Let $B$ and $\tilde{B}$􏰑 be unital point separating subalgebras of ...
10
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0answers
412 views

Witt's proof of Gelfand-Mazur / Ostrowski's Theorem

Previously asked on Math Stackexchange without answers. Background: As sort of a hobby, Ernst Witt gave extremely short proofs for famous theorems. This question is about his six-line proof of the ...
1
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0answers
72 views

norm of operator between matrix algebras equipped with trace norm [duplicate]

‎Let $M_i$ stands‎ ‎for the algebra of $d_i\times d_i$ matrices with $\|T\|=d_i‎ ‎\|T\|_1=d_i (trace{(T^\ast T)}^{\frac{1}{2}})$‎, ‎and $M_{ij}$‎ ‎stands for the algebra of $d_i d_j\times d_i d_j$ ...
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1answer
54 views

How to show $\lambda_i \in \sigma_A(x)$?

Let $\sigma_A(x)$ be the spectrum of $x$ in $A$, and linear functional $\phi$ satisfying $\phi(x)\in \sigma_A(x)$ for every $x \in A$, consider $p(\lambda)=\phi((\lambda e-x)^n)$, and denote its roots ...
4
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1answer
211 views

A precise definition of contractible Banach algebras

I asked this question at MSE but I did not received any answer. So I ask it here at MO I am sorry if this question is elementary: What is a precise definition of a contractible Banach ...
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0answers
211 views

The second dual of $C(X)$ with the compact-open topology

Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
7
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1answer
315 views

Open projections and Murray-von Neumann equivalence

Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
4
votes
1answer
254 views

A generalization of unsolvable equation $ab-ba=1$ in a Banach algebra

It is well known that the equation $$(*)\;\;\;\;ab-ba=1$$ is unsolvable in a Banach algebra. I search for some reasonable generalization of this equation in higher variable for investigation ...
6
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1answer
504 views

Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
0
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0answers
60 views

weakly amenable weighted sequence algebras

Let $v=(v_n)_{n\in\mathbb{N}}$ be a positive weight with $\inf_nv_n>0$ (for convenience we may take $v_n\geqslant1$). Then $\ell_{\infty}(v)$ is a Banach algebra with coordinate-wise multiplication....
1
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1answer
124 views

Does the image of $f$ contain a positive number?

Let $H$ be a Hilbert space and $T$ be a bounded and positive operator on $H$. Define a real function $f$ on positive real numbers by $$f(r):=\|(r+T)^{-1}\|^{-1}-r\quad(r\in\mathbb R_+).$$ Does the ...
2
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1answer
165 views

Regarding spectral radius

Let $A$ be a $C^*$ algebra. Let $a\in A$ be such that $a^*a-aa^*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?
2
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1answer
88 views

A Question about an irreducible ultra-power II,

Let $E$ be an irreducible Banach $A$-module, for a Banach algebra $A$. One can easily show that for an ultra filter $\mathcal U$, $(E)_\mathcal U$ is a Banach $(A)_\mathcal U$-module. Is it possible ...
2
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0answers
54 views

Tensor product and quotients of it

Let $A,B$ be Banach algebras, and $I$ be a closed two sided ideal of $A$ and $J$ be a closed two sided ideal of $B$. Is the relation $A\hat{\otimes}B/I\hat{\otimes}J\cong A/I\hat{\otimes}B/J$ true?(...
3
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1answer
151 views

Proof of $L^1(\mathbb{R}) \ast f \neq L^1(\mathbb{R})$

It is known that $L^1(\mathbb{R}) \ast f$ is dense in $L^1(\mathbb{R})$ for some $f\in L^1(\mathbb{R})$. So for such $f$ the closure of $L^1(\mathbb{R}) \ast f$ in the $L^1$ norm is $L^1(\mathbb{R})$....
5
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1answer
131 views

Are there any non-trivial convergent sequences in the maximal ideal space of the measure algebra?

Consider the measures on the circle, $M(\mathbb T)$, endowed with the convolution product which makes it a unital Banach algebra under the total variation norm. Denote by $\Delta$ the maximal ideal ...
0
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0answers
64 views

A question about an irreducible ultra-power

Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
13
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4answers
367 views

About the existence of characters on $B(X)$

Let $X$ be a Banach space. Let $B(X)$ be the space of all bounded linear operators on $X$. Does $B(X)$ have an empty character space for any $X$? I know the proof of the fact that $M_n(\mathbb{C})$ ...
11
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2answers
695 views

Operator that commutes with projections

We investigate the Hilbert space $\ell^2(\mathbb{N}_0)$ with standard orthonormal basis vectors $e_n:=(0,...,0,1,0,...).$ Consider the family of self-adjoint rank $1$ projections $P_n\bullet:= \...
0
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0answers
51 views

An ideal in Fourier–Stieltjes algebras $B(G)$

Let $G$ be a locally compact group and $R$ be any family of representations of $G$. Let $A_R(G)$ be the closed linear span in Fourier–Stieltjes algebras $B(G)$ of the coefficient functions of all ...
1
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1answer
101 views

Local branch of logarithm in commutative Banach algebras

Assume That $A$ is a commutative complex Banach algebra. Let $G$ be the connected component of invertible elements containing the identity. Is there an smooth embedded curve $c:(-\epsilon, \...
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0answers
82 views

Can we express separability of a ray-remainder in terms of the function algebra?

Let $X = [0, 1)$ be a ray and $C(X)$ the algebra of bounded continuous real functions. The spectrum of $C(X)$ is the Stone-Cech compactification $\beta [0,1) $ of the ray. It's easy to see the ...
4
votes
1answer
115 views

approximate diagonal

Let $I$ be an arbitary index set, $((A_i)_i,\|.\|_i)_{i\in I}$ be a family of Banach algebras, with approximate diagonal $(m^{i}_α)_α\subseteq A_i\hat\otimes A_i$, and $B=\{(x_i)_i\in {\displaystyle \...
0
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1answer
75 views

Projective tensor product

Let $A$ and $B$ be Banach algebras. Then the map $\phi:(A\widehat\otimes A) \oplus_\infty (B\widehat\otimes B) \to (A\oplus_\infty B)\widehat\otimes(A\oplus_\infty B)$ is a contractive embedding. Can ...
0
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1answer
213 views

Is it true that $A$ is Morita equivalent with $M_I(A)$ [closed]

Let $A$ be a unital Banach algebra. Is it true that $A$ is Morita equivalent with $M_I(A)$, where $I$ is an arbitrary index set ($M_I(A)$ is the space of $I*I$ matrices with entries in $A$. Let $a,b\...
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0answers
112 views

Generalization of the Chinese remainder theorem

Let $A$ be a Banach algebra and $\{I_{\alpha}\}_{\alpha}$ be a collection of closed two-sided pairwise coprime ideals of $A$. Is the Chinese remainder theorem true for $A$ and $\{I_{\alpha}\}_{\alpha}$...
3
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1answer
80 views

About Beurling algebras

Do there exist an amenable Beurling algebra that is neither Arens regular nor strongly Arens irregular? In his memoir "The second duals of Beurling algebras", A. T. Lau proved that there exists a ...
0
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1answer
149 views

About the topological center of a Banach algebra

Let $\mathfrak A$ be a Banach algebra with a bounded approximate identity (BAI), and let $\square$ and $\lozenge$ denote, resp., the first and the second Arens products of $\mathfrak A''$. Consider ...
4
votes
1answer
179 views

Bases closed under multiplication

Let us say that a Hamel basis $H$ in an algebra $A$ is closed under multiplication, if $ab\in H$ whenever $a,b\in H$. It is an easy observation that if $A$ has such a basis then there it also has a ...
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0answers
91 views

Decomposition of Banach bimodules of Banach algebras

Let $A$ and $B$ be Banach algebras, $\theta:A\rightarrow \mathbb{C}$ be a character (i.e., a multiplicative linear functional) and $A\oplus _{\theta} B$ be the $l^1$-direct sum of $A$ and $B$ equipped ...
2
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1answer
146 views

Function in $B(\mathbb{R})$

Denote by $B(\mathbb{R})$ the set of all functions on $\mathbb{R}$ which are representable in the form $f(x)=\int_{\mathbb{R}}e^{itx}d\mu(t)$, where $\mu$ is a finite complex-valued Borel measure. ...
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0answers
53 views

One question about $L^1(G/K)$ and its closed subalgebra of $K$-invariant functions $L^1(G)^{\sharp}$

Can someone please clarify explicitly why: "The smallest closed subspace of $L^1(G/K)$ containing $L^1(G/K)^{\sharp}$ and invariant under the (left) $G$-action, is the full space $L^1(G/K)$". Where $...