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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Projections in Tensor Product

Let $X$ and $Y$ be Banach spaces (algebras). If $X\otimes^\pi Y$ denotes the projective tensor product of $X$and $Y$, define $P:X\otimes^\pi Y\rightarrow X$ as follows : for $x\otimes y\in X\otimes ...
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Positive Elements: Characterization

Problem Given a C*-algebra with unit $1\in\mathcal{A}$. Define positive elements as: $$A\geq0\iff\sigma(A)\geq0\quad(A=A^*)$$ Positive elements can be characterized by: $$A\geq0\iff A=B^*B$$ ...
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124 views

Norm on unitisation of a $C^\ast$ algebra

In the theory of $C^\ast$ algebras there exists the following theorem: If $A$ is a $C^\ast$ algebra and $\widetilde{A}$ denotes its unitisation then there exists exactly one norm that extends the ...
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168 views

Positive Operators: Definition?

Definitions Given an operator algebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ with $1\in\mathcal{A}$ Consider selfadjoint operators $A=A^*\in\mathcal{A}$. Define positive elements by: ...
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50 views

In unital Banach algebra $r(a^n) = (r(a))^n$

I tried to prove the following: If $A$ is a unital Banach algebra and $r(a)$ denotes the spectral radius then $r(a^n) = (r(a))^n$. Could somebody please tell me if I got this proof right? Thanks. ...
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23 views

What other ideals are there in this subalgebra of the disk algebra

Let $A$ be the disk algebra and $A_0 = \{f \in A \mid f(0) = 0\}$. I am trying to give an example of a maximal non-modular ideal in $A_0$. I have tried $I=\{f\in A_0 \mid f(1) = 0\}$ and proved that ...
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64 views

Any finite group G is amenable

A linear functional $μ:L^∞ (G)→\mathbb{C}$ is called a mean on $L^∞ (G)$ if $ μ(1)=1$ and is positive, i.e. if $μ(f)≥0$ for all positive $f∈L^∞ (G)$. A group G is amenable if there exists an ...
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62 views

Hermitian adjoint of an isometry

Let $u: H \to H$ be an isometric operator on a Hilbert space. Let $\ast$ be an involution. I was wondering if $u^\ast$ is also an isometry. I tried to prove it but didn't quite manage. Then I ...
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48 views

Clarification for this exercise needed

I would like to solve the following exercise but there are a few minor things I am not clear about: Let $A$ be the Banach algebra of $C^1([0,1])$ endowed with the norm $\|f\|=\|f\|_\infty + ...
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25 views

Maximal abelian subalgebra in algebra of matrices

Let $A$ be the Banach algebra of $n \times n$ matrices over $\mathbb C$. Then the subset consisting of all diagonal matrices is an abelian subalgebra. (correct me if I'm wrong). Now I want to show ...
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194 views

$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.

Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
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34 views

Local Module Homomorphism

Let $A$ be Banach algebra and $X,Y$ be two left Banach $A$-modules. It is said that the linear bounded map $\phi:X\to Y$ is left $A$-module homomorphism if for any $a\in A$ and any $x\in X$ we have ...
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32 views

Uniqueness in Bochner's theorem

Bochner's theorem : Let $G$ be a locally compact Abelian group. Then for any $ \phi \in \ P(G) $ there is a unique positive Radon measure $ \ μ \in \ $ M ($ \widehat{G} $) such that ...
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0answers
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How I can make $\Bbb{C}[x]$ into a Banach algebra?

Let $\Bbb{C}$ the complex field. Define $\Bbb{C}[x]$ as the set of all polynomials with variable $x$. It is known that $\Bbb{C}[x]$ is a algebra. Now the question is this that how I can make it a ...
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63 views

On maximal ideal spaces of a banach algebra

I am reading this article on maximal ideal spaces and there is this part that I don't quite understand very well, hope you guys can help me out. "Let $M(A)$ denote the maximal ideal space of a ...
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1answer
80 views

Identifying the dual group of a locally compact abelian group with the spectrum of $ {L^{1}}(G) $.

Folland stated the following theorem in his book A Course in Abstract Harmonic Analysis on Page 88. The dual group of a locally compact abelian group can be identified with the spectrum of $ ...
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1answer
30 views

projection generated by intersection of two projection

let $H$ be a Hilbert space and $P,Q$ be projections on $H$. suppose $P,Q$ do not commute. $P\wedge Q$ is a projection on $PH\cap QH$. I want to calculate $P\wedge Q$ but I can not. Please help me. ...
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1answer
25 views

Commutative subspace lattice

I have seen an article in which there is an algebra which was named CSL-algebra (Commutative Subspace Lattice). This algebra is about projection on Banach algebra? I couldn't find any good source to ...
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39 views

Locally compact group and continuous function with compact support

Suppose that $G$ is a locally comact group, let $H$ be a open subgroup of $G$ and let $\phi:G\to\Bbb C$ be a continuous function such that $Supp\phi=:\overline{\{x\in G: \phi(x)\neq 0\}}$ is compact. ...
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For which $s\in\mathbb R$, is $H^s(\mathbb T)$ a Banach algebra?

According to Theorem 4.39 in Adams & Fournier Sobolev Spaces: If $mp>n$, $m\in\mathbb N$, then $W^{m,p}(\Omega)$ is a Banach algebra, provided that $\Omega\subset\mathbb R^n$ satisfies the ...
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37 views

Inequality norms

Let $A$ be a bounded linear operator on a Banach space $X$. Can we show that for an arbitrary $n \in \mathbb{N}$ and $x \in X$ such that $\|x\|_X \geq 1$ we have that $$\|A^n x \| \leq \|Ax\|^n.$$ ...
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1answer
33 views

Convergence of the sequence of operators on a Banach space.

Let $T$ be a bounded linear operator on a Banach space $E$, and let $(T_n)$ be a sequence of bounded linear operators on $E$ such that $$\| (T - T_n)x \| \to 0 \ \ \text{ as } n \to \infty$$ for all ...
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70 views

Module homomorphism

Let $A$ be a Banach algebra with norm $\|.\|_A$ and $X$ be a Banach space with norm $\|.\|_X$. If there exists a operation $.:A\times X\to X$ such that for any $a,b\in A$ and $x,y\in X$ we have ...
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1answer
181 views

Example of a Banach algebra $ A $ whose only closed ideals are $ \{ 0 \} $ and $ A $.

I'm trying to come up with an example of a Banach algebra $A$ that is not commutative, unital and such that the only closed ideals are $\{0\}$ and $A$. I already struggled to even come up with a non ...
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1answer
54 views

Module algebras

Spectrum: For Banach algebra $A$ spectrum is denoted by $\sigma(A)$ and defined as the set of all non-zero bounded linear multiplicative function from $A$ to $\Bbb C$.(Function $\psi:A\to\Bbb C$ is ...
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263 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
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71 views

Polynomial Calculus: well defined?

Disclaimer This thread has been refreshed! Problem Given a C*-algebra $\mathcal{A}$. Consider an element $A\in\mathcal{A}$. Introduce an abstract polynomial calculus: $$A=A^*:\quad ...
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1answer
67 views

Polynomial Ring: Root vs. Remainder

Framework: Consider a unital ring: $e\in R$ and a given polynomial: $p\in R[X]$ (Note that I do not require the ring to be an integral domain.) Problem: If it has a root then it factorizes: ...
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1answer
65 views

Spectrum: Polynomials

It is written in Bratteli-Robinson that some simple transformations yield the relations: $$\sigma(a+A)=a+\sigma(A)$$ $$\sigma(A^n)\subseteq\sigma(A)^n$$ The latter one is deduced by the ...
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33 views

Normalized States

A linear functional is normalized iff it preserves identity: $$\|\omega\|=1 \iff \omega(\mathrm{id})=1$$ Can somebody help me proving it? (I just remember it was kind of an easy thing.)
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Is it true that $0<0$ is a part of the proof for a unique fixed point of a contraction map?

I think it isn't, but my friend told me that this statement is true, he said that $0<0$ is even part of the proof for a unique fixed point of a contraction map. I google it but I couldn't find ...
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1answer
31 views

Approximating a mean on $L^{\infty}(G)$ by a norm $1$ non-negative $f\in L^{1}(G)$

Let $G$ be a locally compact group. A mean $M$ on $L^{\infty}(G)$ is a continuous linear functional on $L^{\infty}(G)$ such that $1 = \langle 1 , M\rangle = \|M\|$. My Exercise: Show that the set ...
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1answer
77 views

Convergence Radius: Non-Analyticity

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
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1answer
93 views

Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have ...
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1answer
36 views

Is this subalgebra of a semisimple algebra semisimple?

Let $A$ be a semisimple algebra and $e$ be an idempotent in $A$. Then $eAe=\{eae:a\in A\}$ is a subalgebra of $A$ with $e$ as the identity. We want to prove that $eAe$ is also semisimple. That is, if ...
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Show that $B(X)$ is semisimple for a Banach space $X$ [duplicate]

Show that $B(X)$ is a semisimple Banach algebra, where $X$ is a Banach space. That is, to show that rad $B(X)=\{0\}$, or equivalently, to show $\sigma(AT)={0} \, \forall T\in B(X)\Rightarrow A=0$. I ...
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20 views

How to prove the set of fourier multipliers is a banach algebra?

Hi I am new here at math stack Exchange, this is my first question, hope you guys can help me out:) Suppose $F\colon L^2(\mathbb{R} ) \to L^2(\mathbb{R})$ is the Fourier transform given by ...
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49 views

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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1answer
48 views

Characters of $C(X)$

I showed that the character space $\Omega (\ell^1 (\mathbb Z))$ is homeomorphic to $S^1$. Now I am wondering if there is a similar identification for $C(X)$ where $X$ is compact Hausdorff with the ...
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1answer
56 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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1answer
58 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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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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1answer
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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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89 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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23 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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61 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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38 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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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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121 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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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 ...