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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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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33 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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26 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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33 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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19 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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35 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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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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18 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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66 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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79 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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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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57 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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94 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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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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33 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
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58 views

Centralizers and containment of $c_0$

Let $X$ be a Banach space over $\mathbb{R}$ or $\mathbb{C}$. By a multiplier on $X$ we mean a bounded linear operator $T$ on $X$ such that every extreme point of $B_{X^*}$ becomes an eigenvector for ...
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56 views

How to prove $\widehat{(fg)}(n) = \hat{f} (n) \ast \hat{g} (n); (n \in \mathbb Z)$?

Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows: $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$ It is ...
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32 views

Spectral Radius given by Neumann Series

Good afternoon everybody. So far it is clear that outside the spectral "disk" the resolvent is given by the Neumann Series. But why does it tell us that there is a point on the spectral "circle" where ...
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19 views

How to factorize exponential as a convolution of finite number of functions( series)?

We choose a points $x_{1},..., x_{n}$ in $\mathbb Z$ (not 0), such that $$x_{k+1}\not = x_{1}+x_{2}+...+x_{k} \ \ (k=1,2,...,n-1).$$ Let $\delta_{x}$ denote the measure of total mass $1$, ...
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How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
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35 views

Does there exists $f\in A(\mathbb T)$ such that $||f||=r$ and $||\mathrm{e}^{if}||= \mathrm{e}^{r}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t)\, \mathrm{e}^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb ...
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27 views

Continuity of additive maps

I studied all of previous posts about "when an additive map is continuous?" but I did not get my answer! My question is the following: Let $f:A\longrightarrow B$, be an bijective map from a Banach ...
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35 views

How to inter change of norm and limit in the Banach algebra?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
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Properties of an additive mappings which preserves projections

Let $A$ and $B$ be two $C^{*}$-algebras and $\Phi:A\longrightarrow B$ be an additive map which satisfies $\Phi(0)=0$, $\Phi(I)=I$ and $\Phi$ preserves projections, (i.e, $\Phi(P)=Q$ where $Q$ is also ...
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65 views

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 $\mathcal{A}^{-1}_0$ ...
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66 views

Do the homomorphisms really have to be continuous?

I read that If $\varphi, \psi$ are continuous homomorphisms from a normed algebra $A$ to a normed algebra $B$ then $\varphi = \psi$ if $\varphi$ and $\psi$ are equal on a set $S$ that generates the ...
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28 views

Show that a subalgebra is commutative.

If $B$ is an unital algebra (even not commutative), how do I show that the subalgebra spanned by the elements $1$, $f$ and $(f - \lambda1)^{-1}$ is commutative? Thank you.
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Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
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94 views

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 ...
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36 views

Coset of the Abstract index group of a Banach Algebra?

I'm studying on the book of Douglas: "Banach algebra techniques in operator theory" and there is a passage I don't understand, and I hope you can give me a hand. "A continuous function $f$ from $X$ ...
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37 views

Injective norm on tensor algebra of a finite-dimensional Banach space

Suppose that $V$ is a finite n-dimensional Banach space, and suppose that $T(V)$ is the tensor algebra on it. Furthermore, suppose that $T^{(n)}(V)$ is the "abridged" tensor algebra obtained by taking ...
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$\|1-f\| < 1 \Rightarrow \exp(\log(f))=f$ if $f$ is in a Banach Algebra.

I'd like to know if it's possible to prove the statement without passing by theorems about power series and convergence radius. The series are as usual: $$\exp(x) = \sum_{n=0}^\infty ...
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Does there exists $f\in L^{1}(\mathbb R)\cap FL^{1}(\mathbb R)$(=Fourier algebra) but $|f|\not \in A(\mathbb R)$?

For $f\in L^{1}(\mathbb R)$; We define the Fourier transform of $f$ as follows: $$\hat{f}(\xi)= \int_{\mathbb R}f(x) e^{-2\pi i \xi x}dx; \ \text {for} \ \xi \in \mathbb R.$$ Consider a Fourier ...
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1answer
99 views

Is the product rule true in a Banach algebra?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the ...
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70 views

Why is the character space of a Banach algebra weak $\ast$ closed?

Let $A$ denote a unital commutative Banach algebra and let $\Sigma(A)$ denote its character space. Why is $\Sigma(A) \cup \{0\}$ weak $\ast$ closed in the closed unit ball of $A^\ast$? Is $\Sigma(A) ...
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1answer
93 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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37 views

Limit of the spectrum in Banach algebra

Let $A$ an unital complex Banach algebra, $a_{n} $ is a sequence such that $\lim_{n\to \infty}a_{n}=a$. What is the relation between $\lim_{n\to \infty}\sigma(a_{n})$ and $\sigma(a)$. I think that it ...
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64 views

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 ...
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1answer
113 views

The disk algebra and continuous homomorphisms

The disk algebra is the set of continuous functions $f: D \to \mathbb C$ where $D$ is the closed unit disc in $\mathbb C$ and $f$ is analytic on the interior of $D$. It is endowed with the ...
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1answer
74 views

Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
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318 views

The disk algebra and continuous functions

The disk algebra is the set of continuous functions $f: D \to \mathbb C$ where $D$ is the closed unit disc in $\mathbb C$ and $f$ is analytic on the interior of $D$. It is endowed with the ...
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what is a “Banach algebra” without the norm condition on a continuous multiplication?

I wish to use the following finite-dimensional Banach spaces. although they do not need to be Banach algebras for my proposed application, a mild curiosity is aroused, because the multiplication ...
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90 views

Ideals and exactness of projective tensor product of Banach spaces / algebras

Thanks for suggesting this question: Image of the tensor product of strict maps of Banach spaces I read the reference and realize that for a short exact sequence of Banach algebra: $0 \to J \to A \to ...
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1answer
26 views

$\|(I+A)^{-1}\| \leq \frac{1}{1-\|A\|)}$

I have the following problem, of which I have a slight problem to finish with the second part: Let $X$ be a Banach space and let $A \in B(X)$, $\|A\| < 1$. Prove that $(I+A)^{-1}$ exists and is ...
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2answers
69 views

$\ell^1(\mathbb{Z})$ 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 ...
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1answer
42 views

Prove that the Gelfand transform $\widehat{f}$ is uniform algebra

I'm going to find an example of uniform algebra and show that satisfying the definition. Example: Show that The Gelfand transform $\widehat{f}$ is uniform algebra. We know that: A uniform ...
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1answer
54 views

How do we show that prime C* algebras have trivial center

A prime C* algebra is a C* algebra with the property that the product of any two of its non zero ideals is non zero. The claim is that it has trivial center, i.e., the only central elements are ...
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65 views

Uniform boundedness

I was thinking about the following problem: Let $(A_t)_{t \geq 0}$ be a family of bounded operators on a Banach space $X$ which is uniformly bounded and let $(B_{t,\alpha})$ be a net in the Banach ...
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48 views

Question on representation of a Banach algebra

Let $A$ be a Banach algebra and $\pi$ a continuous irreducible representation of $A$ on a Banach space $X$. Suppose $\pi(a)\xi\neq0$ for some $a\in A$ and some $\xi\in X$. Let $\eta\in X$. The ...
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68 views

Norm of inverse, Banach algebra, Proof of Gelfand-Mazur theorem

In a proof of the Gelfand-Mazur theorem, I read that $$ \lim_{\lambda\to\infty}\|(a-\lambda 1)^{-1}\|=0 $$ (where it is assumed for the sake of contradiction that the inverses in the norm exist for ...