0
votes
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 ...
1
vote
1answer
40 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 ...
2
votes
1answer
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 ...
1
vote
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. ...
0
votes
1answer
46 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) \} ...
1
vote
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
1answer
37 views

Unitization of Banach algebras

Is every theorem about unital Banach algebra also true for non-unital Banach algebras because of unitization?
1
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1answer
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 ...
1
vote
1answer
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 ...
1
vote
0answers
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 ...
0
votes
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 $\tau \in A^\ast$: ...
0
votes
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 ...
4
votes
1answer
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 ...
4
votes
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 ...
2
votes
0answers
20 views

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 ...
2
votes
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 ...
2
votes
0answers
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 ...
0
votes
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 $T\in A(E)$. ...
1
vote
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 ...
3
votes
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 ...
0
votes
0answers
52 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 ...
0
votes
0answers
26 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 ...
0
votes
0answers
17 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 $1$, ...
1
vote
1answer
32 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 ...
0
votes
1answer
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 ...
4
votes
1answer
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 ...
1
vote
0answers
29 views

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 ...
0
votes
1answer
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 ...
0
votes
1answer
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) ...
0
votes
1answer
77 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 ...
4
votes
1answer
55 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 ...
2
votes
1answer
66 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 ...
1
vote
1answer
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 ...
1
vote
1answer
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 ...
2
votes
1answer
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 ...
1
vote
1answer
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 ...
4
votes
1answer
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 ...
2
votes
0answers
112 views

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: ...
0
votes
1answer
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 ...
0
votes
1answer
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 ...
1
vote
1answer
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 ...
1
vote
1answer
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 ...
6
votes
1answer
112 views

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) ...
1
vote
1answer
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. ...
4
votes
1answer
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 ...
1
vote
1answer
37 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 $\bar{z}$ ...
0
votes
2answers
71 views

Limit of nth power of operator norm

I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies ...
2
votes
1answer
98 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...