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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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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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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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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23 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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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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24 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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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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27 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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24 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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80 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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60 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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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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36 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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55 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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73 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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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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277 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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64 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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25 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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62 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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40 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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39 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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48 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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43 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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36 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 ...
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93 views

Maximal ideals in function spaces

Hello friends of mathematics, i have got a question. In the lecture we proved that the maximal ideals of $C(X)$ are the sets of functions which vanishes on a closed subset of $X$. But now i will look ...
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30 views

Why is this quotient algebra contractible? (Blackadar 21.4.3)

I ran into this question when reading the proof of Theorem 21.4.3 in Bruce Blackadar, K-theory for Operator Algebras The question is as follows: Given $A$, $B$ Banach algebras, and a surjective ...
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Homeomorphism to the unit interval $I$

my task is to find the maximal ideal space of the semisimple Banach algebra $\mathfrak{B}=C^1[0,1]$. Can someone give me a hint? I read that this should be homoemorphic to the uni interval $[0,1]$, ...
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Functional analysis problem involving maximal ideal space

For the past week, I've been trying to solve, as a practice homework exercise, Problem 6 of Chapter 11 in Rudin's Functional Analysis, but have not gotten very far it seems. The problem is as follows: ...
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Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
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ultrapowers of a matrix

1)Let A, B, C, D be a Banach algebras, and U be a free ultrafilter. Can we see that ultrapowers of \begin{pmatrix} A & B \\ C & D \end{pmatrix} equal to \begin{pmatrix} (A)_U & ...
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61 views

Square root in Banach algebra

Suppose we are given a unital Banach algebra $A$ and an element $a\in A$ such that the spectrum is a subset of the positive reals $\mathbb{R}_{>0}$. Then by a theorem (see for example W. Rudin ...
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72 views

Continuity of a map from $\mathbb{C}$ to a Banach algebra

Consider the map from $\mathbb{C}$ to a unital Banach algebra $B$ given by $x \mapsto \exp(xb)$ for $b\in B$. I proved that this map is continuous by using the definition of $\exp(xb)$ as a contour ...
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45 views

Integration of rational function on Banach algebra

I do not follow the proof of this Theorem Theorem Suppose$R(\lambda) = P(\lambda) + \sum_{m,k}c_{m,k}(\lambda - \alpha_m)^{-k}$ is a rational function with poles at the points $\alpha_m$. ($P$ ...
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37 views

Minkowski type inequality in Banach algebras

Under which circumstances it is true that $\|(A+B)^n\|^{1/n}\le \|A^n\|^{1/n}+\|B^n\|^{1/n}$ for elements $A$ and $B$ in a Banach algebra and a natural number $n$?
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87 views

Show existence of a continuous function with certain properties

Let $X$ be a compact Hausdorff space and $C(X)$ the commutative algebra of continuous complex-valued functions endowed with the maximum-norm. Let $J\subset C(X)$ be an ideal and $g\in J$. Let ...
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146 views

Banach algebra problem: $\left\|e^{ta}\right\|\leq Me^{-\omega t}$

Let $A$ be a unital algebra, and $a\in A$. Assume that $\sigma(a)\subset \{\lambda\in \mathbb{C}: Re\lambda < 0\}$. Show there exists $M,\omega >0$ such that $$\left\|e^{ta}\right\|\leq ...
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46 views

Matrix-valued function

I have a problem about matrix-valued function. Given a function $f:\mathbb{R}^k \rightarrow {\cal M}_{k \times k}$ of class $C^1$, where ${\cal M}_{k \times k}$ is the set of all $k \times k$ ...
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Can $f*g = f+g$ for $f$ and $g$ compactly supported?

Let $f$ and $g$ be continuous, compactly-supported functions $\mathbb{R} \to \mathbb{C}$. Can it happen that $f*g = f+g$? Here, $f*g$ denotes the convolution $$(f*g)(s) = \int_\mathbb{R} f(t) g(s-t) ...
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42 views

In what kind of Banach algebras is 0 the only topological zero divisor?

On page 33 of http://math.aalto.fi/opetus/harmanal/pruju/calg04.pdf it is asked in what kind of Banach algebras is 0 the only topological zero divisor. What do they mean by kind of Banach algebras. ...
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35 views

Spectrum in Banach Algebra Example

What's answer: Let $A=l^\infty(S)$, where S is a non-empty set. Then $\sigma(f)=\overline{(f(S))}$ for all $f\in A$. It's mention in Murphy as example (1.2.2).
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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 $\phi(a-b+I)=\phi(I)=0$. ...
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106 views

In banach algebra: Show there exists $x$ s.t. $x^2+bx+xb+c=0$

Let $A$ be a banach algebra with unit $e$ and $b,c\in A$ be such that $\sigma(b^2-c)\subset \mathbb{R}^+$ Then there exists $x\in A$ such that $$x^2+bx+xb+c=0$$ I want to show this, but lack the ...
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36 views

Is holomorphic functions on (0, 1) (vanishing at endpoint) dense in $C_0((0, 1))$?

Here is my argument, please let me know if it works or not. By Stone-Weierstrass Theorem (Complex Version), functions in $C_0((0, 1))$ can be uniformly approximated by polynomials in z and $\bar{z}$ ...