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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Prove that if $A$ is algebra generated by $\sin(x)$ and $\cos(x)$ then $A = \{ f\in C_b(\mathbb R ) : f(t) = f(t + 2\pi )$ for all $t \in \mathbb R\}$ [duplicate]

Possible Duplicate: Finding a closed subalgebra generated by functions. Let $A$ be the uniformly closed subalgebra of $C_b(\mathbb{R} )$ generated by $\sin(x)$ and $\cos(x)$. Prove: $A = ...
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Subadditive, submultiplicative and scalar-multiplication-invariant functions

Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, ...
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186 views

Boolean algebras of projections

Suppose $X$ is a Banach space with an unconditional basis $(e_n)$. Then, one may easily define a Boolean algebra of projections in $\mathcal{L}(E)$ which is isomorphic to the power-set of $\mathbb{N}$ ...
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Ideals in Commutative Banach Algebras

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 ...
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Boundary of invertibles in a normed algebra

A student and I are reading the book Introduction to Banach Spaces and Algebras, by Allan, and we're stuck. Exercise 4.5 says: Let $A$ be a normed algebra with unit sphere $S$. Let $a\in A$. ...
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300 views

Normed algebra with multiplicative norm, non-isomorphic to $R$, $C$, $H$.

Is there an infinite dimensional real normed algebra $A$ such that $\|xy\|=\|x\|\cdot \|y\|$ for all $x,y \in A$? Thanks.
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187 views

Finding a closed subalgebra generated by functions.

Consider the space of all bounded continuous real-valued functions of $\mathbb{R}$. I am having trouble understanding how to find the closed subalgebra generated by sine and cosine.
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Involutive and C* Banach Algebras.

I want to prove the next theorem: If $\pi: A \rightarrow B$ is a star homomorphism, meaning it's an algebra homomorphism which also satisfies: $\pi(x^*)=(\pi(x))^*$, where $A$ is an involutive Banach ...
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Can one deduce Liouville's theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?

As you probably know, the classical proof of the non-emptiness of the spectrum for an element $x$ in a general Banach algebra over $\mathbb{C}$ can be proven quite easily using Liouville's theorem in ...
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270 views

Reflexive Banach algebras?

I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces. For a Banach space $X$, we investigate its dual $X'$ ...
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426 views

Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor ...
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262 views

a question about invertibility of Banach Algebra

If $X$ is a Banach algebra with identity, and $0$ is the only element $x \in X$ such that there is a sequence $\{ {x_n}\} \subset X$, $\left\| {{x_n}} \right\| = 1$ and $x{x_n} \to 0$ or ${x_n}x \to ...
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251 views

How far is a Banach algebra from being a multiplicative group?

Given a Banach algebra $\mathcal{A}$, the collection of invertible elements in $\mathcal A$, $G(\mathcal{A})$ is a group. I wonder whether there is a measurement for how far $\mathcal{A}$ is from ...
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213 views

Why is $GL(B)$ a Banach Lie Group?

Banach Lie Groups are what you'd expect: http://www.encyclopediaofmath.org/index.php/Lie_group,_Banach If $B$ is a Banach algebra then why is $GL(B)$, the set of invertible elements of $B$, a ...
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632 views

Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
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196 views

Does the Banach algebra $L^1(\mathbb{R})$ have zero divisors?

Assume that the functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are integrable and equal to zero on $(-\infty,0)$, (i.e $f,g \in L^+$). Then by Titchmarsh's theorem: $f*g$ is zero almost everywhere ...
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Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
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126 views

Do non-commutative algebras with dense commutative subalgebras exist?

Let $A$ be a normed unital algebra. Suppose that $C\subseteq A$ is a commutative subalgebra which is dense in $A$. I ask myself the following question: Under the above assumptions, is $A$ necessarily ...
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185 views

When is the ring of continuous functions absolutely flat?

This question was created in a discussion. Let $X$ be a topological space. Denote by $C(X; \mathbb{R})$ the ring of real-valued continuous functions defined on $X.$ Characterize those compact ...
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334 views

How to prove Halmos’s Inequality

How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from ...
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Reference request: Fourier and Fourier-Stieltjes algebras

I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...
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The identity cannot be a commutator in a Banach algebra?

The Wikipedia article on Banach algebras claims, without a proof or reference, that there does not exist a (unital) Banach algebra $B$ and elements $x, y \in B$ such that $xy - yx = 1$. This is ...
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Abelian sub-C*-algebras

Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the ...
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114 views

Geometric Sums in Banach Algebra

Let $E$ be a Banach Algebra with identity, and $v\in E$, so that $||v|| < 1$. The geometric series $w = \sum_{k=0}^\infty v^k$ converges in the norm. I can show that $||w|| \le ...
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388 views

The Wiener algebra question

I want to show that the maximal ideal space of the Wiener algebra $W$ is $ \{ M_z : z \in \mathbb{T} \}$ where $M_z = \{ g \in W : g(z)=0 \}$ Could you please help me?
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2answers
625 views

Is there an algebraic homomorphism between two Banach algebras which is not continuous?

According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces. Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...
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3answers
669 views

Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
4
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2answers
297 views

If $(I-T)^{-1}$ exists, can it always be written in a series representation?

If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$. Thinking in terms of a ...
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227 views

When is the weighted space $\ell^p(\mathbb{Z},\omega)$ a Banach algebra ($p>1$)?

Let $\omega:\mathbb{Z}\to (0,\infty)$ and let $1\leq p<\infty$. Consider the space $\ell^p(\mathbb{Z},\omega)$ of complex valued sequences $f=(a_n)_{n \in \mathbb{Z}}$ such that ...
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295 views

Converse of Wiener's lemma

Let $A$ be a commutative Banach algebra with unit. It is well known that if the Gelfand transform $\hat{x}$ of $x\in A$ is non-zero, then $x$ is invertible in $A$ (the so called Wiener Lemma in the ...