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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31
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1k views

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 ...
12
votes
2answers
668 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 ...
9
votes
1answer
444 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 ...
6
votes
1answer
444 views

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.
11
votes
1answer
687 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 $\mathcal I$ ...
3
votes
1answer
363 views

strictly positive elements in $C^*$-algebra

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to find the following:a)What are the strictly ...
4
votes
2answers
310 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 ...
4
votes
1answer
227 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 ...
8
votes
2answers
241 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 ...
6
votes
1answer
201 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.
6
votes
3answers
729 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 ...
1
vote
2answers
219 views

$Conv(Ex((C(X))_1))$ is dense in $(C(X))_1$?

Let $X$ be compact Hausdorff, and $C(X)$ the space of continuous functions over $X$. Denote the closed unit ball in $C(X)$ by $(C(X))_1$, then it can be shown $f$ is an extreme point of $(C(X))_1$ if ...
8
votes
1answer
242 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 ...
7
votes
1answer
232 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 ...
6
votes
1answer
237 views

On the spectrum of the sum of two commuting elements in a Banach algebra

Original: Soit A une algèbre de Banach unitaire et a et b deux éléments tels que a*b=b*a. Pourquoi σ (a+b) с σ(a)+σ(b) Et qu’elle est la relation entre σ (a*b) et σ(a) et σ(b)? Translation: Let ...
4
votes
1answer
564 views

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 ...
9
votes
1answer
424 views

Liouville's theorem for Banach spaces without the Hahn-Banach theorem?

Let $B$ be a (complex) Banach space. A function $f : \mathbb{C} \to B$ is holomorphic if $\lim_{w \to z} \frac{f(w) - f(z)}{w - z}$ exists for all $z$, just as in the ordinary case where $B = ...
3
votes
2answers
203 views

Left topological zero-divisors in Banach algebras.

Let $ A $ be a unital Banach algebra. Define $ \zeta: A \longrightarrow [0,\infty) $ by $$ \forall a \in A: \quad \zeta(a) \stackrel{\text{def}}{=} \inf_{b \in \mathbb{S}(A)} \| ab \|, $$ where $ ...
6
votes
2answers
183 views

Stone-Čech via $C_b(X)\cong C(\beta X)$

I am having some trouble constructing the Stone-Čech compactification of a locally compact Hausdorff space $X$ using theory of $C^*$-algebras. I did some search but could not find a good answer on ...
5
votes
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.
5
votes
1answer
78 views

Why locally compact in the Gelfand representation?

I'm missing something in the Gelfand representation. Let's just say $\mathfrak{A}$ is a Banach algebra. Then it's a Banach space, and so we have $\mathfrak{A}^\ast$. The multiplicative linear ...
3
votes
2answers
256 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 ...
2
votes
1answer
33 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 ...
1
vote
1answer
158 views

‎If ‎‎$‎X$ is an infinite-dimensional Banach space and ‎‎$‎‎u‎\in ‎B(X)‎$ ,then $\bigcap_{v\in K(X)}\sigma(u+v) =\cdots$

‎If ‎‎$‎X$ is an infinite-dimensional Banach space and ‎‎$‎‎u‎\in ‎B(X)‎$,why the following equality is true? $$\bigcap_{v\in K(X)}\sigma(u+v) =\sigma(u) \setminus \{\lambda \in\mathbb{C}\mid u - ...
7
votes
1answer
327 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.
4
votes
2answers
75 views

Characterization of small Banach subalgebras

Let $A$ be a unital Banach algebra and $x \in A$ nonzero. We can consider the subalgebra $B$ of $A$ generated by $\{1,x\}$. This is the norm closure of the subspace of polynomials in $x$. So for any ...
3
votes
1answer
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 ...
3
votes
0answers
130 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
3
votes
2answers
133 views

If $a\ge 0$ and $b\ge 0$, then $\sigma(ab)\subset\mathbb{R}^+$.

This is an exercise in Murphy's book: Let $A$ be a unital $C^*$-algebra and $a,b$ are positive elements in $A$. Then $\sigma(ab)\subset\mathbb{R}^+$. The problem would be trivial if the algebra ...
2
votes
1answer
54 views

How do we show that prime C* algebras have trivial center

A prime C* algebra is a C* algebra with the property that the product of any two of its non zero ideals is non zero. The claim is that it has trivial center, i.e., the only central elements are ...
2
votes
1answer
268 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 ...
1
vote
0answers
25 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 ...
1
vote
2answers
52 views

Definition of subalgebra

Is it generally implicitly assumed that if $B$ is a subalgebra of a unital Banach algebra $A$ then $1 \in B$? I tried to find a definition of subalgebra but the only definition I found was in Murphy ...
1
vote
2answers
93 views

Disk algebra norm clousre

I have a trouble with a question and i need help to solve it. Define $A_1$={$f$ $\in C(\overline{\mathbb{D}})$ | f is analytic in $\mathbb{D}\}$ $A_2$=the norm closure of polynomials in ...
0
votes
1answer
109 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 ...
0
votes
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 ...
0
votes
3answers
99 views

The maximal ideal space of $A$ is contained in the unit ball of $A^\ast$

On page 281 of Rudin's book Functional Analysis, let $\Delta$ be the maximal ideal space of a commutative Banach algebra $A$, $K$ be the norm-closed unit ball of $A^*$, then $\Delta\subset K$ (by ...
-1
votes
1answer
51 views

what is a “Banach algebra” without the norm condition on a continuous multiplication?

I wish to use the following finite-dimensional Banach spaces. although they do not need to be Banach algebras for my proposed application, a mild curiosity is aroused, because the multiplication ...