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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When do Fourier series and Fourier transform coincide

The other day I proved that if $f \in \ell^1 (\mathbb Z)$ then its Gelfand transform $\widehat{f}$ is a map $S^1 \to S^1$ such that $$ \widehat{f}(z) = \sum_{k \in \mathbb Z}f(k) z^k$$ and that ...
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Characters of $\ell^p(\mathbb Z)$

The other day I showed that the character space $\Omega (\ell^1 (\mathbb Z))$ is homeomorphic to $S^1$. Now I am wondering if there are similar identifications for $\ell^p(\mathbb Z)$ when $p \in ...
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15 views

Gelfand transform on disk algebra

I tried to prove that if $A$ is the disk algebra then the Gelfand transform is the identity map. The statement can be found here in Theorem 4.4 but it is given without proof. Please can someone check ...
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1answer
46 views

Gelfand transform is a bijection between $\ell^1$ and $\mathbb D$?

Let $A=\ell^1 (\mathbb Z)$. I read that it is possible to identify $S^1$ with the character space $ \Omega (A)$. But I have constructed a proof that identifies $ \Omega (A)$ with $\mathbb D$, the ...
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42 views

Character space of $L^{1} (\mathbb Z)$

I have a question about the Gelfand and norm topologies on the character space of $L^{1} (\mathbb Z)$. Are the Gelfand and norm topologies equal, on the character space of $L^{1} (\mathbb ...
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32 views

What are these spectra (part 1)

I needed some insight into spectra in Banach algebras. To this end I tried to calculate a few examples. I will post them not all in one post. Could somebody please tell me if they are correct? Here ...
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35 views

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 ...
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36 views

If composition with a linear functional is continuous, is the function continuous?

If $G$ is an open subset of $\mathbb{C}$ and $f:G \to X$ is a function such that for each $x^*$ in $X^*$, $x^*\circ f:G\to\mathbb{C}$ is analytic, then f is analytic. Will the statement still hold ...
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1answer
63 views

Need help understanding this proof about Gelfand spectrum

Consider the following theorem: Let $A$ be a complex non-unital commutative Banach algebra and let $\Omega (A)$ denote its Gelfand spectrum / character space. Then $\Omega (A)$ is locally compact. ...
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1answer
14 views

Showing the $C^*$ identity

I'm working through a proof in Dixmier's book on $C^*$-algebras and I'm stuck on part of a proof. I'm given a Banach algebra $\mathcal{A}$ which has norm $\lVert\cdot\rVert$ and a semi-norm ...
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45 views

Spectrum and characters: could anyone please check my proof

I tried to prove the following: Let $A$ be a commutative non-unital complex Banach algebra and $\chi : A \to \mathbb C$ a character. Then $$ \sigma (a) = \{\chi (a) : \chi \in \Omega (A) \} ...
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27 views

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 ...
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78 views

Why characters are continuous

According to Wikipedia: ''Every character is automatically continuous from $A$  to $\mathbb C$, since the kernel of a character is a maximal ideal, which is closed. '' where $A$ is a Banach algebra. ...
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30 views

The space of all bounded sequences over a Banach Algebra.

If $A$ is a commutative Banach algebra, can we identify $l^\infty(A)$ as the dual of $ l_1\hat\otimes A$ (the projective tensor product of $l_1$ with $A$? Also, what do we know about the quotient ...
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40 views

Need help understanding this proof

I need help understanding the following: If $A$ is a (complex) banach algebra and $I$ is a proper modular ideal then $\overline{I}$ is also proper. Proof. Let $u\in A$ be such that $a-ua, a-au \in ...
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51 views

Is it a typo in the statement of this theorem

Consider the following theorem: If $I$ is a modular maximal ideal of a unital abelian algebra $A$, then $A/I$ is a field. It is a basic fact of algebra that if $R$ is a commutative unital ring then ...
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30 views

Need some help finishing this proof about characters in Banach algebras

I tried to prove: Let $A$ be a commutative unital complex Banach algebra. Then there is a bijection between the maximal ideals in $A$ and the set of non-zero homomorphisms $A \to \mathbb C$. But I ...
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1answer
37 views

Unitization of Banach algebras

Is every theorem about unital Banach algebra also true for non-unital Banach algebras because of unitization?
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38 views

Number of isomorphisms between two fields

Let $F,F'$ be two fields. Is there anything that can be said about the number of isomorphisms that can exist? In particular can there be more than one? What if $F$ is the complex numbers $\mathbb C$? ...
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17 views

Characters in Banach algebras

I am reading Wikipedia and there is something I don't understand: ''Let $A $ be a unital commutative Banach algebra over $\mathbb C$. Since $A $ is then a commutative ring with unit, every ...
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1answer
88 views

On the mean value theorem in $\mathbb R^2$

Consider the following claim: If $A$ is a (complex) unital Banach algebra and $f: \mathbb R \to A$ is differentiable with $f' = 0$ then $f$ is constant. The proof uses that for $\tau \in A^\ast$: ...
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26 views

What are these spectra (part 2)

I needed some insight into spectra in Banach algebras. To this end I tried to calculate a few examples. I will post them not all in one post. Could somebody please tell me if they are correct? Here ...
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1answer
281 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
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1answer
41 views

How are these subalgbras

This question was prompted by the following example: If $X$ is compact then $C(X)$ is a Banach algebra and if $U$ is an open subset of $X$ then $C_0(U)$ is a subalgebra of $C(X)$. Here $C(X)$ is ...
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Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
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31 views

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 ...
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2answers
44 views

Definition of subalgebra

Is it generally implicitly assumed that if $B$ is a subalgebra of a unital Banach algebra $A$ then $1 \in B$? I tried to find a definition of subalgebra but the only definition I found was in Murphy ...
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35 views

What about $\ell^1$ with pointwise multiplication

This question occurred to me after reading this thread. I was working on finding an example of a Banach algebra. The example I came up with was $\ell^1 (\mathbb N)$ with pointwise multiplication. I ...
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8 views

Spectrum in infinite hilbert space

Show that if a complex hilbert space $H$ is separable, then for every compact set $K$, there exists a bounded linear operator $T:H→H$ such that $σ(T)=K$S
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1answer
41 views

Spectrum of idempotent element

Let $A$ be some unitary algebra over $\mathbb{C}$. If $a^2=a$ and $0\ne a\ne 1$ then $\{0,1\}\subset \sigma_A(a)$ ($\sigma_A(a)$ is the spectrum if $a$). I believe that also $\sigma_A(a)\subset ...
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87 views

Two dimensional Banach algebras

I have this one to solve: Let $A_1$ be an matrix algebra of matrices in form $ \pmatrix{ \alpha & 0 \\ 0 & \beta\\ }$ and $A_2$ be an matrix algebra of matrices in form $ \pmatrix{ \alpha ...
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Can we expect, $\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$ in the Banach algebra $A(\mathbb R)$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$ and consider Fourier algebra $$A(\mathbb R):= \{f\in ...
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How to use $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in Banach sapace $C([0, T];M^{p,1})$?

(For details of this question you may see the paper,(proof of theorem 1, page.9), MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; Bull. Lond. Math. Soc. 41 ...
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19 views

Functional Calculus in C*-algebra

I came across this function while studying functional calculus: for $0<\alpha<1$, define $$s^\alpha= \int \frac{1}{1+st}t^\alpha \frac{dt}{t}$$ Now suppose we define $a^\alpha$ for some normal ...
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1answer
20 views

Classification of Banach Algebras?

Is there a classification theorem for Banach algebras, or even for Banach *algebras, similar to the GNS representation theorem for $C^*$-algebras? If yes, please provide a reference where I can read ...
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1answer
16 views

Quotients of Banach algebras by ideals

I am currently working through Banach Algebra Techniques in Operator Theory and am hung up on a detail on 2.32. When trying to show that the quotient of a Banach space $\mathcal{B}$ by a closed ideal ...
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1answer
30 views

Why is $F_\phi$ defined on the whole disk

This is a question about a proof on page 97 in these lecture notes. In exercise 13, I don't understand On the hand, $F_\phi$ is defined on the whole open disk $D$ Why is $F_\phi$ defined on ...
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1answer
18 views

Set difference of spectra

Let $A$ be a $B^{*}$ algebra and let $B$ be a sub $B^{*}$ algebra.From the fact that $d({\sigma}_{B}(x))$ is a subset of $d({\sigma}_{A}(x))$ where $d$ is the boundary., deduce that ...
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1answer
25 views

On the spectral radius $r(ab) $

Let $A$ be a unital complex Banach algebra and $a,b\in A$. Define $r(a) = \sup_{\lambda \in \sigma(a)} |\lambda|$ where $\sigma(a)$ denotes the spectrum of $a$. Note that $\sigma (ab) \setminus ...
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29 views

Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
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1answer
75 views

Concatenation with continuous function is entire

Apologies. I have to ask two questions in one and I will give you the reason below. The questions are: If $f$ is entire and $g$ is continuous does it follow that $g\circ f$ is entire? If ...
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1answer
92 views

Are these two steps in this proof necessary?

Theorem: Let $A$ be a unital Banach algebra. Then for $a \in A$ the spectrum $\sigma (a) \neq \varnothing$. Consider the following proof: The first step that seems unnecessary to me: Let's say we ...
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1answer
14 views

SOT Convergence and Compact Convergence

Let $E$ be a Banach space, and let $A(E)$ denote the closure of the finite rank opertors on $E$. Let $(S_\alpha)$ be a bounded net of operators on $E$ such that $S_\alpha T\to T$ for all $T\in A(E)$. ...
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1answer
31 views

Non-modular maximal ideal in abelian Banach algebra

Let $A$ be the disk algebra (i.e. the algebra of all functions that are continuous on the closed unit disk and analytic on the open unit disk) and let $A_{0}=\{f\in A:f(0)=0\}$. Then $A_{0}$ is a ...
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1answer
73 views

Is taking limits in this proof really necessary?

Consider the following proof: Theorem. Let $A$ be a unital Banach algebra and $a$ an element of $A$ such that $\Vert a \Vert < 1$. Then $1 - a \in \operatorname{Inv}(A)$ and $$(1-a)^{-1} = ...
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1answer
51 views

A closed ideal in a commutative Banach algebra $C(X)$

Suppose that $A$ is a natural Banach function algebra on $K$, a compact Hausdorff space. So $A$ is realised as an algebra of continuous functions on $K$, is a Banach algebra for some norm (necessarily ...
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63 views

boundary of a spectrum proof

Let $A$ be a closed unital subalgebra of banach algebra $B$. Prove that ${\delta}{\sigma}_{B}(x)$ is contained in ${\delta}{\sigma}_{A}(x)$ for every $x$ in $B$.
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12 views

banach norms which are not equivalent on semisimple banach algebra

we know that on every semisimple commutative banach algebra all banach norms are equivalent. but if i drop semisimple or commutative will it work? if not please give examples of banach norms which are ...
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51 views

homomorphism or not

Let $T$ be a bounded operator on $H$ and fix a vector $x\in H$. Define $f$ on the space of polynomials in $T$ by $f(p(T))=p(x)$. Is $f$ a homomorphism? Initally I thought it obvious but the subtelty ...
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30 views

Gelfand Transform in a specific case

What is the gelfand transform of an operator in the algebra generated by a bounded normal operator and it's adjoint? Thanks