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Questions tagged [banach-algebras]

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that |xy| ≤ |x||y|. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use (c-star-algebras) or (von-neumann-algebras) instead (or in addition). Further related tags: (operator-algebras), (operator-theory), (banach-spaces), (hilbert-spaces).

A Banach algebra is an algebra over the real or complex numbers which is equipped with a complete norm such that $|xy| ≤ |x||y|$. The study of Banach algebras is a major topic in functional analysis. If you are about to ask a question on C*-algebras or von Neumann algebras please use or instead (or in addition). Further related tags: , , , .

1330 questions
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Gelfand transform is an isometry

I'm having a bit of trouble showing that the Gelfand transform $A \rightarrow C(\operatorname{sp}(A))$ is isometric iff $\|x^2\| = \|x\|^2$ for a general unital commutative Banach algebra. For a $C^*$ algebra, where we know $xx^*$ is closed in $A$…
Ben Stott
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Finding a closed subalgebra generated by functions.

Consider the space of all bounded continuous real-valued functions of $\mathbb{R}$. I am having trouble understanding how to find the closed subalgebra generated by sine and cosine.
josh
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Given a Banach Algebra $A$ and elements $x,y \in A$. If $x$ and $xy$ are invertible, then so is $y$.

A silly question, but I don't see the answer. This question is from Rudin's Functional Analysis, Chapter 10, Exercise 1a). It's obvious that $y$ has a left inverse as $((xy)^{-1}x)y=e$, where $e$ is the unit of the algebra. However, starting with…
Malcolm King
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Invertible elements in a Banach algebra connected to identity

I'm currently working on a problem for homework in my Banach algebras course and I've run into a bit of an issue with terminology. Let $\mathcal{A}$ be a Banach algebra, then $\mathcal{A}^{-1}_0$ denotes the subgroup of invertible elements that can…
Cameron Williams
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Disk algebra norm clousre

I have a trouble with a question and i need help to solve it. Define $A_1$={$f$ $\in C(\overline{\mathbb{D}})$ | f is analytic in $\mathbb{D}\}$ $A_2$=the norm closure of polynomials in $C(\overline{\mathbb{D}})$ i need to show that $A_1$=$A_2$ i…
david
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Continuity of product in Banach Algebras

EDIT: The definition literally says that the product is bilinear, as Chris Eagle kindly pointed out. Turns out that my original proof was fine. This question does no longer require an answer and therefore can be closed (if that's appropriate). Thank…
commie trivial
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Topological zero divisors of $\mathbb C^n$

We know that every zero divisor is a topological zero divisor but not every topological zero divisor is a zero divisor. First I define the terms: Zero divisor: In a Banach Algebra $A$ an element $x\in A$ is said to be a zero divisor if there exist…
Shreya Chauhan
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Let $K$ be a circle. Describe the spectra of two subalgebras of $C(K)$

Suppose $K=\{\lambda\in\mathbb{C}: 1<\vert\lambda\vert<2\};$ put $f(\lambda)=\lambda$. Let $A$ be the smallest closed subalgebra of $C(K$) that contains $1$ and $f$. Let $B$ be the smallest closed subalgebra of $C(K$) that contains $f$ and…
Matema Tika
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Are compact operators trace class operators?

We say that $A\in B(\mathcal{H})$ is a trace class operator, if $\sum_{i\in I}\langle|A|e_i,e_i\rangle<\infty$,$\hspace{0.1cm}$ such that {$e_i; i\in I$} is a orthonormal bass for Hilbert space $\mathcal{H}$. If compact operator on Hilbert space…
Alireza
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Condition in the definition of Banach star algebra

Here the definition of Banach star algebra is given as Banach algebra with an involution. In the book by Murphy for example, it is given as Banach algebra with an involution plus the condition that $\|a\|=\|a^\ast\|$. My question is: is the…
user167889
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Riesz Projection in Functional Analysis.

By definition the Riesz projection of a Banach algebra element $a$ associated with a complex number $\alpha$ is given by $p(\alpha , a)= \frac{1}{2\pi i} \int_{\Gamma} ( \mu -a)^{-1} \,\, \text{d}\mu,$ where $\Gamma$ is a small curve isolating…
MrD
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If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element of Banach algebra $\mathcal{A}$ with unit $e$.

I was reading an article yesterday which was silent on the algebra of Banach. In that article was provided this example If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element of Banach algebra $\mathcal{A}$ with unit $e$.…
user145717
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Why is the kernel a maximal ideal

Assume $A$ is a commutative unital Banach algebra and $\tau : A \to \mathbb C$ is a character. I can prove that $I = \mathrm{ker}(\tau)$ is a maximal ideal using some basic abstract aglebra. The problem is, there is an alternative argument that I do…
Student
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$\ell^1(\mathbb{Z})$ with discrete convolution and the Gelfand transform

I want to show the following: $\ell^1(\mathbb{Z})$ with the discrete convolution $*$ is isomorphic to a subalgebra of $C(T)$ where $T = \{z \in \mathbb{C} : |z| = 1\}$. The following theorem seems to be handy, Let $\Lambda$ be the maximal…
guest
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Show that there is a unique continuous function

I have no idea where to even start, i have never dealt with question like this before, any direction you can give me would be greatly appreciated.
greg
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