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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37
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2k 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 ...
4
votes
1answer
650 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 ...
12
votes
2answers
921 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 ...
12
votes
1answer
1k 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 $\mathcal I$ ...
10
votes
1answer
539 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 ...
8
votes
3answers
307 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 ...
6
votes
1answer
586 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.
5
votes
2answers
328 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
325 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 ...
9
votes
1answer
417 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
852 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 ...
5
votes
1answer
274 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.
8
votes
2answers
305 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 ...
3
votes
1answer
80 views

Prove a condition for a Banach algebra to be isometrically isomorphic to $\mathbb C$

Can anyone help me by providing a detailed verification of the following theorem? Let $\mathcal{A}$ be a Banach algebra. If there exists $M<+\infty$ so that $$\Vert a \Vert\Vert b \Vert\leq M ...
6
votes
3answers
1k 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 ...
5
votes
1answer
191 views

Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
2
votes
2answers
59 views

If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element of Banach algebra $\mathcal{A}$ with unit $e$.

I was reading an article yesterday which was silent on the algebra of Banach. In that article was provided this example If $a\in\mathcal {A},$ and $\Vert a\Vert <1$, then $e-a$ is regular element ...
1
vote
2answers
241 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 ...
0
votes
1answer
55 views

$C_{c}(X)$ is complete. then implies that $X$ is compact. [closed]

Let $X$ is locally compact Hausdorff space .If $C_{c}(X)$ is complet,then $X$ is compact (this is to be proved). I know that $C_{c}(X)$ is dense in $C_{0}(X)$. As $C_{c}(X)$ is complete implies that ...
8
votes
1answer
270 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 ...
10
votes
1answer
713 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 = ...
6
votes
1answer
420 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 ...
3
votes
2answers
288 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 $ ...
2
votes
1answer
128 views

Extensions: Spectrum

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1'\in\mathcal{A}'$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
7
votes
2answers
284 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
60 views

On the spectrum of a product in a Banach algebra, in specific case

Let $A$ be a Banach algebra, and suppose that $a,b\in A$ have spectra that satisfy: $\sigma(a) \subset U$, and $\sigma(b)\subset U$, where $U$ is the open right half-plane of complex numbers with ...
5
votes
1answer
112 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 ...
5
votes
1answer
579 views

Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
3
votes
2answers
271 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
0answers
162 views

Semigroups: Entire Elements (I)

Problem Given a Banach space $E$. Consider a C0-semigroup: $$T:\mathbb{R}_+\to\mathcal{B}(E)$$ Define its generator by: $$Ax:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)\in E$$ (It is a densely-defined ...
2
votes
1answer
78 views

Does *-operator be automatically continuous

In the C*-algebras, does the * -operator be automatically continuous? I think it is yes, because C*-algebras are semisimple, from the Johnson Theorem, it must be automatically continuous. Am I right? ...
1
vote
1answer
250 views

Maximal ideals in function spaces

Hello friends of mathematics, i have got a question. In the lecture we proved that the maximal ideals of $C(X)$ are the sets of functions which vanishes on a closed subset of $X$. But now i will look ...
0
votes
1answer
53 views

What's difference between spectrum and eigenvectors of an operator

Let $x$ be an operator in $B(H)$. By definition $\sigma(x)=\{\lambda \in \Bbb C ~; \lambda - x \neq inv \}$. Also to find eigenvalue of an operator we should find $\lambda$ such that $x\xi = \lambda ...
7
votes
1answer
392 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.
5
votes
1answer
176 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.
4
votes
2answers
88 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
110 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
196 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
161 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
36 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 ...
2
votes
1answer
94 views

Nonhermitian elements whose spectra are real?

Let $\mathcal{A}$ be a complex C* algebra. It is well-known that $\sigma(a)\subset\mathbb{R}$ if $a$ is a hermitian. I wonder whether the converse is true. That is, if $\sigma(a)\subset\mathbb{R}$, ...
2
votes
1answer
293 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
1answer
160 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 ...
1
vote
1answer
183 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 - ...
0
votes
2answers
66 views

Function Spaces: Characterizations of Positivity

Context The problem here is about the characterization of positivity for real or complex valued functions: $$\sigma(f)\geq 0\iff\sigma(f(x))\geq 0\text{ for all }x\in X\iff f(x)\geq 0\text{ for all ...
0
votes
1answer
148 views

Is the product rule true in a Banach algebra?

Let $X$ be a Banach space and $\mathcal{L}(X)$ the Banach algebra of all bounded linear operators $L:X\to X$, where the norm is given by $$\|L\|_\mathcal{L}=\sup\{\|L(x)\|_X;\;\|x\|_X=1\}$$ and the ...
0
votes
1answer
64 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 ...
0
votes
1answer
41 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 ...
5
votes
1answer
97 views

K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
3
votes
3answers
87 views

Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.

Consider $H$ as a Hilbert space. How can I find a Banach space $E$, for that, $H=B(E)$ where $B(E)$ is the set of bounded linear operator on $E$? (At least under some conditions on $H$) Also if the ...