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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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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69 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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1answer
74 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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52 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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47 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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90 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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1answer
147 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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51 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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128 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) ...
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47 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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39 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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43 views

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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1answer
107 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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1answer
44 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 $\bar{z}$ ...
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1answer
27 views

Show that there is a unique continuous function

I have no idea where to even start, i have never dealt with question like this before, any direction you can give me would be greatly appreciated.
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1answer
31 views

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 ...
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1answer
60 views

Density of linear span of idempotents in $L^{\infty}$

How do I show that the linear span of idempotents is dense in $L^{\infty}(\Omega,\mu)$ where $(\Omega,\mu)$ is a measure space? I don't really have any idea how to do this. Does it involve ...
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2answers
90 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 ...
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1answer
106 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 ...
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1answer
47 views

(Commutative Banach algebra) Prove that $G(\mathcal A)$ be an open set in $\mathcal A$.

UPDATE I have a problem: Let $\mathcal A$ be a commutative Banach algebra. Denote $G(\mathcal A)$ is the set of all invertible elements in $\mathcal A$. Prove the following assertions: ...
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33 views

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. ...
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223 views

Banach Algebra: $\sigma(xy)\cup\{0\} = \sigma(yx)\cup\{0\}$

It is Rudin excercise 10.4 where we aim to prove $\sigma(xy)\cup\{0\} = \sigma(yx)\cup \{0\}$ for elements $x,y\in A$ a Banach-algebra.( $\sigma$ being the spectrum) In (a) we prove that $e-yx$ ...
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70 views

$L(\ell_{p})$ contains only one proper closed ideal

I am trying to solve the following problem: Show that if $1<p<\infty$ and $T:\ell_{p}\rightarrow\ell_{p}$ is not compact then there is a complemented infinite dimensional subspace $E$ of ...
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1answer
99 views

If a field contains the complex field, then it is $\mathbb{C}$

This question is originated from a book by Gaal, (Linear Analysis and Representation Theory). Theorem 7 from section 6, chapter 1 reads as follows, and I quote: "Theorem 7: Let $A$ be a complex, ...
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1answer
48 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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56 views

Do inclusions of Banach algebras preserve spectral radius?

Let $f : A_1 \to A_2$ be an injective homomorphism of unital Banach algebras. It's a standard fact that if $f$ is has closed range, i.e. $A_1$ is embedded as a closed subalgebra of $A_2$, then for ...
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1answer
36 views

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 ...
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129 views

A question on multiplicative linear functional on Banach algebra.

I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume $A$ is a non-unital C*-algebra and $\tilde{A}$ is its unitization (the elements of the form ...
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64 views

Banach algebra.

Iam new in this field. I am reading a paper and have encoutered the following Lemma. Let $u\in F_{1}.$ Then $Sp(u)=\{0, tr(u)\},$ where $F_{1}$ is the set of one-dimensional elements and tr(u) is the ...
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31 views

Diagonalizing operator over $L^2(\mathbb{T})$

I've been asked to diagonalise an operator on $L^2(\mathbb{T})$, given by $Tf(z) = f(z^{-1}$). I know that I'm expected to find a $U$ such that $TU = UM_f$, where $M_f$ is the multiplication operator, ...
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76 views

Representation of subspaces as complemented subspaces

Let $X$ be any separable Banach space. The Banach-Mazur theorem states (astonishingly) that $X$ is isometric to a closed subspace of $C(\Delta)$, the space of continuous functions on the cantor set ...
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Positive elements in a C*-algebra [closed]

Prove that if $a$ is an element in a $C^*$-algebra $A$, then $a$ is positive if and only if $f(a) \geq 0$ for every state $f$ on $A$.
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15 views

Closure of the set of fredholm perturbation

Let $A$ and $B$ be two unital Banach algebras and $T\colon A\to B$ an homomorphism of Banach algebras. Let denote the set of Fredholm perturbation elements in $A$, i.e. $\operatorname{Ft}:=\{r\in ...
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232 views

Wiener's theorem in $\mathbb{R}^n$

Reading Stein's "Singular integrals and differentiability properties of functions" I came across the following statement (this is in the proof of Lemma 3.2, pages 133-134): We now invoke the ...
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1answer
104 views

Gelfand transform is an isometry

I'm having a bit of trouble showing that the Gelfand transform $A \rightarrow C(\operatorname{sp}(A))$ is isometric iff $\|x^2\| = \|x\|^2$ for a general unital commutative Banach algebra. For a $C^*$ ...
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42 views

Definite positive measure and GNS representation

Let $G$ be a locally compact group. Let $\mu$ be a positive definite complex measure ([D, p295]): we have $\mu(f*f^*)\geq 0$ for any compact support continuons function $f \in C_c(G)$. In [D, p ...
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59 views

Is power of convergence nets, convergence?

Let $A$ be a Banach algebra and $(f_{\alpha})$ is a net in $A$ and convergence in norm to $f$. Is $(f^{n}_{\alpha})$ convergence in norm to $f^{n}$ for every $n$ in $\mathbb{N}$?
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1answer
111 views

Example of a singular element which is not a topological divisor of zero

We know that every topological divisor of zero in a commutative Banach algebra is singular. I need an example of a singular element which is not a topological divisor of zero.
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47 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 ...
4
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122 views

Invertibility of elements in a Banach algebra

Let $X=L^1\cap L^2$, and $\hat{X}$ be the Banach algebra of the image under Fourier transform of $X$. Then do the unital extension $1\dot{+}\hat{X}$ of $X$ by adding a constant function with the norm ...
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1answer
160 views

Uniqueness of the involution on a $C^*$-algebra

indication please Let $A$ be a C*-algebra. Suppose that there exists on $A$ another involution $x\rightarrow x^{\#}$ such that $||xx^{\#}||=||x||^2$, for all $x\in A$. Prove that $x^{\ast}=x^{\#}$, ...
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1answer
58 views

Roots in Banach algebras.

I'm studying positive functionals on normed algebras and I got stuck in the following problem: Let $A$ be a unital Banach algebra, and $x\in A$ be such that $\Vert x\Vert <1$. Then the series ...
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1answer
36 views

A question about Banach algebras: showing that $\operatorname{Sp}a \subset D_o \cup D_1$

Maybe this problem be easy for a person that have study in Banach Algebra; please give me a hint. Let $e=0$ or $1$, and $a$ be an arbitrary element in a Banach algebra $A$. Let $D_o$ and $D_1$ be the ...
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71 views

Commutative Noetherian Banach algebra.

Prove that: 1) Every commutative real unital Noetherian Banach algebra with no zero divisors is isomorphic to the real or complex numbers. 2) Every commutative real unital Noetherian Banach algebra ...
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118 views

Commutative unital Banach algebra with nilpotent elements

What would be a concrete example of a commutative unital Banach algebra that contains nilpotent elements?
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1answer
105 views

Show that: If $A$ is an arbitrary abelian Banach algebra, its spectrum is totally disconnected

I don't know how should I start to show: If $A$ is an arbitrary abelian Banach algebra in which the idempotents have dense linear span, its specrum (the space of characters on $A$) is totally ...
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1answer
111 views

Prove the approximate identity from the unitization

Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge (or monotonous converge) to $1$ in $A^+$, does $\{x_n\}$ must be the ...
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1answer
361 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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3answers
527 views

Banach Algebra counterexample

Can someone give me an exemple of a Banach Algebra $\mathbb{A}$, for which there is no isometric representation in a Hilbert Space ? (with proof or references to proof) Thank you very much :)
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1answer
115 views

Bergman-Shilov Boundary and Peak Points

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by ...