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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2answers
58 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
3
votes
1answer
83 views

Spectrum of normal elements in C*-algebras

Let $\mathcal{A}$ be a C*-algebra and $x \in \mathcal{A}$ a normal element. Can you show that $\left\{ \phi(x) : \phi \text{ is a state on } \mathcal{A} \right\}$ is the closed convex hull of the ...
2
votes
1answer
45 views

C*-algebra representations

Let A be a C*-algebra and $\left\{\phi_n\right\}$ a weak* dense sequence in the state space. Putting $\phi=\sum_n 2^{-n} \phi_n$, can you show that $\phi$ is a state and the representation $\pi_\phi$ ...
3
votes
1answer
48 views

$\int_{\mathbb R^{2}} |\int_{\mathbb R} (f(t-y)- f(t)) g(t-x) e^{-2\pi i w\cdot t} dt|dx dw \to 0 $ as $y\to 0$?

Fact: It is well-known that translation is continuous in the $L^{1}$ norm, that is, if $f\in L^{1}(\mathbb R)$ then $\lim_{y\to 0} \|f_{y}-f\|_{L^{1}(\mathbb R)}=0;$ (where, $f_{y}(x)= f(x-y)$, ...
3
votes
1answer
118 views

How to prove $F$ is a contraction mapping?

Given a $n\times n$ matrix $\mathbf{D}$ in which each entry $\mathbf{D}_{ij} \in [0,1]$. Let $i$-th row vector of $\mathbf{D}$ denote by $\mathbf{D}_{i\ast}$. The mapping $F:[0,1]^n \mapsto [0,1]^n$, ...
1
vote
0answers
28 views

For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
9
votes
1answer
314 views

Maximal ideals and maximal subspaces of normed algebras

This is a kind of "prove or give a counter-example" question, and I'm having some difficults with it: By a maximal ideal $I$ of an algebra $A$, we mean an ideal $I\neq A$ which is not properly ...
0
votes
0answers
30 views

Follow up on star algebra (proof verification)

I previously asked this question about a proof of the following claim: If $A$ is a commutative non-unital non-zero $C^\ast$ algebra then $\Omega (A)$ is not empty. In the meantime I believe to ...
0
votes
1answer
36 views

Where is commutativity of $b$ needed?

I have a question about the following proof: If $e^{ia}-e^{i\lambda}=(a-\lambda)be^{i\lambda}$ and $(a-\lambda)$ is not invertible then $(a-\lambda)x$ is not invertible for all $x$. Why "since $b$ ...
1
vote
1answer
28 views

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 ...
0
votes
1answer
74 views

C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
2
votes
1answer
113 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 ...
4
votes
1answer
114 views

Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
1
vote
1answer
42 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. ...
0
votes
1answer
21 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 ...
1
vote
1answer
55 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 ...
0
votes
0answers
27 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 ...
0
votes
1answer
47 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 + ...
0
votes
1answer
21 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 ...
6
votes
1answer
182 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 ...
2
votes
1answer
30 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 ...
1
vote
0answers
28 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 ...
2
votes
0answers
38 views

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 ...
4
votes
0answers
43 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 ...
2
votes
1answer
77 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 $ ...
2
votes
1answer
19 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. ...
2
votes
1answer
19 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 ...
0
votes
1answer
38 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. ...
7
votes
3answers
148 views

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 ...
0
votes
0answers
28 views

On the Spectral radius $r(a+b)$ in Banach algebra [duplicate]

Let $\mathcal{L}$ be a complex Banach algebra with a unit. The spectral radius of $x\in\mathcal{L}$ is denoted as $r(x)$. Suppose $a,b\in\mathcal{L}$ and $ab=ba$, how to prove that $$r(a+b)\le ...
1
vote
1answer
32 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 $$\|A^n x \| \leq \|Ax\|^n.$$ ...
1
vote
1answer
24 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 ...
3
votes
1answer
65 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 ...
2
votes
1answer
113 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 ...
2
votes
1answer
49 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 ...
5
votes
1answer
178 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. ...
-1
votes
1answer
45 views

Polynomial Calculus on Spectrum: well defined?

Consider a bounded operator over a Banach space: $T\in\mathcal{B}(E)$ Apply polynomial calculus on the the chosen operator: $p(T),p\in\mathbb{C}[X]$ Why do we need to prove that when two polynomials ...
0
votes
1answer
55 views

Root of polynomial implies vanishing remainder. Application to spectral theory!

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: ...
1
vote
1answer
55 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 ...
0
votes
1answer
30 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.)
0
votes
1answer
66 views

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 ...
2
votes
1answer
28 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 ...
2
votes
1answer
73 views

Convergence Radius => Nonanalytic

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 ...
6
votes
1answer
46 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 ...
0
votes
1answer
31 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 ...
1
vote
0answers
25 views

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 ...
1
vote
0answers
15 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 ...
1
vote
0answers
26 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 ...
1
vote
1answer
40 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 ...
0
votes
1answer
52 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 ...