# Tagged Questions

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) ...

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### application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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### Algebra $A$ and its Gelfand spectrum

Let $A$ be the set of all function $f$ on $\mathbb{R}$ of the form $$f(x)=d+\int\limits_{0}^{\infty}e^{ixt}k(t)dt,\qquad\quad x\in\mathbb{R},$$ where $d\in\mathbb{C}$ and $k\in L_1([0,\infty])$. The ...
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### Show that the Gelfand transform is a morphism?

Let $A$ be a commutative Banach algebra and let $x \in A$. We define the Gelfan transform of $x$ by $$\hat{x} (\chi)= \chi (x)$$ where $\chi$ is a nonzero multiplicative linear functional on $A$. I ...
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### About the Banach algebra $\ell^{\infty} (K( \ell^{1}))$

If $P_{n}: \ell^{1} \rightarrow \ell^{1}$ is the projection onto the first n coordinates, then it's well known that $P_{n} K(\ell^{1}) P_{n}$ is isomorphic to $B(\ell^{1}_{n})$, the Banach space of ...
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### Gelfand spectrum of $l^1(\mathbb Z)$ is homeomorphic to $\mathbb T =\{z \in \mathbb C : |z|=1\}$ [on hold]

Show that the Gelfand spectrum of $l^1(\mathbb Z)$ is homeomorphic to $\mathbb T =\{z \in \mathbb C : |z|=1\}$. Please provide some hints. Thanks in advance.
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### R.Douglas “Banach Algebra Technique Operator Theory” - Chapter 2 issue

Just before 2.37 Corollary (Spectral Mapping Theorem) Douglas says: If $\varphi (z)= \sum_{n=0}^\infty a_nz^n$ is an entire function with complex coefficients and $f$ is an element of the Banach ...
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### If $L(E)$ is a prime C$^*$-algebra?

We say that a C$^*$-algebra is prime if and only if whenever $I$ and $J$ are closed two-sided ideals in $A$ and $I ∩ J = {0}$, then either $I = {0}~ or~ J = {0}$. Let $L(E)$ be the set of all ...
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### Injectivity of the evaluation map from holomorphic functions to a Banach algebra

In Functional Analysis, we have covered Functional Calculus, that is, a way to associate, once having fixed a Banach algebra $A$ and an element $a\in A$, an element $\tilde f(a)\in A$ to every $f$ ...
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### What are the Hermitian idempotent matrices with respect to $l_{1}$ norm

Let $A=(M^{n}(\mathbb{C}), \|\cdot\|_{1})$. The numerical range of $a\in A$ is defined as $V(a)=\{f(a):f\in A',\|f\|=1=f(1)\}$. $a\in A$ is said to be Hermitian if $V(a)\subseteq \mathbb{R}$. $a\in A$ ...
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### Why if $X \subset A$ has codimension $1$, the only subspace properly containing $X$ is $A$?

I saw the following question a while ago on Math.SE This question. The answer provided seems to give a satisfactory result, but one thing in the answer I can not quite see. The author of the answer ...
I am stuck in a problem of Conway's A course in a Functional Analysis. Can anyone give me a hint to solve the problem? The question is "If $A$ is a Banach Algebra, then show that the function $r:A\to ... 1answer 38 views ### The second isomorphism theorem for C*-Algebras in my functional analysis class right now we are studying the basics of C* Algebras and I was recently asked this question about the second isomorphism theorem for C* Algebras, but first let me cite ... 0answers 58 views ### Compactum of Banach algebra I need an example of Banach algebra$A$and a left non-trivial closed ideal$I$with all of following properties: There exists a bounded approximate identity in$I$for$I$i.e., a net$\{e_\alpha\}\...
I want to solve this problem: Let $A$ be an unital Banach algebra and $a\in A$ then r(a)<1 \Longrightarrow \lim_{n\to \infty} a^n=0 \Longrightarrow (1-a)^{-1}=\sum_{n=0}^\infty ...