# Questions tagged [banach-algebras]

The banach-algebras tag has no usage guidance.

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### 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 ...

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### 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 ...

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### 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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### 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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### 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 ...

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### 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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### 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:...

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### 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 ...

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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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### 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*?

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### 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., ...

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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?
...

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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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### 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 ...

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### 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 ...

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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 ...

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### 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 &...

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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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### 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 ...

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### 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 ...

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### 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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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 ...

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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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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 ...

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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 $\...

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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 ...

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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 (\...

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### 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....

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### 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 ...

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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 $\|a\|$?

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### 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 ...

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### 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?(...

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### 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 $L^1(\mathbb{R})$....

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### 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 ...

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### 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 ...

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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 $M_n(\mathbb{C})$ ...

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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:= \...

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### 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
...

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### 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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### 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 ...

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### 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 \...

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### 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 ...

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### 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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### 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 $\{I_{\alpha}\}_{\alpha}$...

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### 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 ...

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### 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 ...

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### 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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### 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 ...

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### 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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### 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 $...