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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: , , , .

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Let $e_0$ be minimal projection, why $E_{e_0}$ is a Hilbert space?

‎A nonzero element $e \in \mathcal{K}(\mathcal{H})$ is called a projection‎, ‎if it is‎ ‎self adjoint and idempotent‎. ‎In addition‎, ‎if $e\mathcal{K}(\mathcal{H})e = \mathbb{C}e$ then‎, ‎it is called a minimal projection‎. ‎Suppose that $e_0\in…
Alireza
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Rudin functional Analysis chapter $10$, exercise $13$

This is Rudin's functional Analysis chapter $10$, exercise $13$. I am confused about the notation $\sigma_A(f)$, what does that mean?(What role does the subscript $A$ play here). And can someone illustrate how to solve this question? Update:…
Toad Jiang
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Ideals of the operator algebra

Let $A$ be a Banach algebra. Is there any relation ship between two-sided closed ideals of $A$ and two-sided closed ideals of the operator algebra $\mathscr B(A)$? Is there any characterization for ideals of $\mathscr B(A)$?
Zeinab
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Is this true? If ‎$‎\varphi\in ‎\mathcal{B}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection. Why?

Is this true? If ‎$‎\varphi\in ‎\mathcal{B}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection, such that $E$ and $F$ are Banach space and $\mathcal{B}(E,F)‎$‎ is the set of all bounded linear operator from $E$ to $F$.
Alireza
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Explaining the theorem

By searching this url http://www2.math.ou.edu/~cremling/teaching/lecturenotes/fa-new/ln7.pdf in google given Theorem 7.4. and given its proff, but I do not understand very well this proof because it is not probably the best written proff think. I…
user145717
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Maximal abelian subalgebra in algebra of matrices

Let $A$ be the Banach algebra of $n \times n$ matrices over $\mathbb C$. Then the subset consisting of all diagonal matrices is an abelian subalgebra. (correct me if I'm wrong). Now I want to show that it is in fact maximal. (I am quite sure it is…
Student
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How to prove $\Omega (A)$ is weak star closed

If $A$ is a unital complex commutative Banach algebra to show that the Gelfand spectrum $\Omega (A)$ is weak star closed how to finish the following arguemnt: My idea was to consider $\tau_n \in \Omega (A)$ such that $\tau_n \to \tau $ pointwise…
Student
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Where is $b-\lambda \notin \mathrm{Inv}(A)$ used in this proof

If $A$ is a unital Banach algebra and $B$ is a closed subalgebra and $\sigma$ denotes the spectra then the following inclusion holds: $$ \partial \sigma_B (b) \subseteq \partial \sigma_A (b)$$ for every $b \in B$. Consider the following proof of the…
Student
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How to prove this map is differentiable

Let $A$ be a unital Banach algebra and $f: Inv(A) \to A$ be the map $a \mapsto a^{-1}$. I'm trying to show that $f$ is differentiable. My idea is to show that the limit of $\delta \to 0$ of $$ {\|(a + \delta a)^{-1} - a^{-1}\| \over |\delta|…
tom b.
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Algebra homomorphism on quotient algebra

Let $\cal A$ be a Banach algebra and $I$ be a closed ideal on it. Let $\phi: \cal A/I\to \cal A$, $\phi(a+I)=a$. Is $\phi$ well defined? if $a+I=b+I$ then $a-b\in I$, so $\phi(a-b+I)=\phi(I)=0$. Since $\phi(a-b+I)=a-b$, $a-b=0$ and so $a=b$…
Albert harold
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Inclusion $[0,1]\rightarrow\mathbb{C}$ generates $C^{1}[0,1]$ as a Banach algebra

I am trying to show that the inclusion map $x:[0,1]\rightarrow\mathbb{C}$ generates $A=C^{1}[0,1]$ as a Banach algebra. The first thing that occurred to me was to try using Stone-Weierstrass but somehow I'm not getting it. Also, I want to show that…
cyc
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Identity in Banach Algebras

This is an extract from Douglas: "Banach Algebra Techniques in operator theory". "For Banach algebras and, in particular, for $C(X)$ the importan idea is that of multiplicative linear functional. [...] Except for the zero functional, which is…
Benzio
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Help me please proving the theorem

I am reading the book of Walter Rudin Functional Analysis and the page 235 was given theorem 10:12 which is as follows: Theorem: If $A$ is a Banach algebra, then $G(A)$ is a open subset of $A$, and the mapping $x\rightarrow x^{-1}$ is a…
Madrit Zhaku
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Representation of a closed ideal in $L^1(T)$.

Let $T$ be the unit circle. Given that $I$ is a closed ideal in $L^1(T)$. I conjecture that there exists a set $S \subseteq \mathbb{Z}$ such that \begin{equation} I=\{f \in L^1(T) : \widehat{f}(n)=0, n \in S\}. \end{equation} What I have shown so…
KK Kwok
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Checking whether $\mathbb C^n$ is semisimple or not

Consider $\mathbb C^n$ as a Banach algebra over $\mathbb C$. Is it semisimple? A Banach algebra is semisimple if the intersection of all maximal ideals of it is ${0}$. I can't figure out what are the maximal ideals of $\mathbb C^n$. How do I solve…
Shreya Chauhan
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