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Questions tagged [banach-algebras]

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) or (von-neumann-algebras) instead (or in addition). Further related tags: (operator-algebras), (operator-theory), (banach-spaces), (hilbert-spaces).

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64 views

$||\phi_z - \phi_w||_{A(\mathbb{D})^*}= 2$ if and only if $z$ or $w$ in $\mathbb{T}$

Let $\mathbb{D}$ be the open unit disc in the complex plane $\mathbb{C}$. Let $A(\mathbb{D})$ stand for the space of all continuous functions on $\overline{\mathbb{D}}$ which are holomorphic in $\...
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11 views

Factorization in a Banach Algebra

Let $ A $ be a Banach algebra. Definition: $ A $ is said to Factor if every $ a \in A $ can be written in the form $ a = bc $ for some $ b, c \in A $. Definition: $ A $ is said to Weakly Factor if ...
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1answer
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A problem with notation on B. Maccluer's book

I'm studying Banach algebras and I'm confused by the following statement: Let $\mathcal{A,B}$ be Banach algebras with common identity and suppose $\mathcal{B}\subset\mathcal{A}$ What does that ...
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1answer
31 views

Maximal modular ideal

I met with some troubles with the two concepts:maximal ideal and maximal modular ideal in $C^*$ algebras. If $I$ is a maximal modular ideal in a $C^*$ algebra $A$,does this imply that for any other ...
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0answers
31 views

Spectrum of the difference of almost commuting elements

Suppose $A$ is a C*-algebra. Suppose we have an approximate unit $(u_\lambda)$ then one knows that $|| u_\lambda a-au_\lambda||\rightarrow 0$ in particular $u_\lambda$ almost commutes with $a$ can ...
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1answer
45 views

Commutative unital Banach algebra not isomorphic to $C(X)$ for some compact Hausdorff space $X$

According to some lecture notes I am reading, "it is not so difficult to find an example of a commutative unital Banach algebra which is not isomorphic to $C(X)$ for some compact Hausdorff space $X$".....
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In amenable Banach algebra if Jacobson radical be finite dimensional then must be semisimple

Let $\mathcal{A}$ be an amenable Banach algebra, I'd like to prove if Jacobson radical rad$\mathcal{A}$ be finite dimensional, then rad$\mathcal{A}$ must be semisimple I know in an amenable ...
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1answer
25 views

Example of a topological space $X$ such that $C_0 (X)$ is not a $C^*$-sub-algebra of $C^b (X)$

Let $X$ be an arbitrary topological space. If $X$ is locally compact and Hausdorff, then $C_0 (X)$ (space of continuous functions vanishing at infinity) is a $C^*$-sub-algebra of $C^b (X)$ (space of ...
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1answer
40 views

“Passing Through Convex Combinations” - Constructing a Norm Convergent Net from a Weakly Convergent Net

Let $ \mathfrak{A} $ be a Banach algebra (or Banach space, I don't think it really matters) and let $ (x_{\alpha}) \subset \mathfrak{A} $ and $ x \in \mathfrak{A} $ be such that: $$ \mathrm{wk} - \...
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1answer
38 views

In Banach *-algebra, every positive linear functional is bounded.

While proving this in class, Our Prof. did it into two cases: One case was if Banach *-algebra has the unit $u$. then he used the following result in his proof: "if $A$ is Banach *-algebra with unit $...
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1answer
41 views

Gelfand-Mazur complex part mean what it says?

This question pertains to the complex Gelfand/Mazur theorem. Mazur/Gelfand says normed division algebra over the complex numbers is isomorphic to the complex numbers. Therefore, is the statement true: ...
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On “An elementary proof of a fundamental theorem in the theory of Banach algebras” by C.E.Rickart

I am reading "An elementary proof of a fundamental theorem in the theory of Banach algebras" by C.E.Rickart, available for instance on https://projecteuclid.org/euclid.mmj/1028998012 The concepts of ...
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37 views

When is a surjective homomorphism between two unital Banach algebras bounded?

Let $A$ and $B$ be unital Banach algebras and $\theta : A\to B$ a surjective homomorphism between these two spaces. What is a sufficient requirement for $\theta$ being bounded, and how would a proof (...
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1answer
22 views

Regarding an entire function being affine

I have been reading this article A characterization of multiplicative linear functionals in Banach algebras and got stuck in the middle of the proof of theorem 1.2 on page 217. In the 3rd line from ...
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1answer
79 views

Using “formal” formulas to get non-formal results

So-called "formal" operations--like "formal differentiation", "formal integration", etc.--have always made me a bit uneasy, because it seems to be used sometimes as a snake-oil solution for dealing ...
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1answer
14 views

Limit of a convergent sequence of invertible elements in a Banach algebra

Question: Let $\{A_{n}\}$ be a sequence of invertible elements in a Banach algebra $\mathfrak{A}$ and suppose $\{A_{n}\}$ has a limit A that commutes with each $A_{n}$. Let $r(A_{n}^{-1})$ denote the ...
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1answer
44 views

Spectrum invariance under the passage to a sub Banach Algebra

Let $\mathcal{B}$ be a unital Banach Algebra, fix $A \in \mathcal{B}$. $\sigma_\mathcal{B}(A) = \{ \lambda \vert \lambda I - A \, not \,invertible \, in \, \mathcal{B}\}$ the specturm of $A$ in $\...
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1answer
15 views

exponential null operator

In $A$ banach algebra with unit, and $X\in A$. if i define $e^X=\sum_{n=0}^{\infty} \frac{1}{n!}X^n$ why $e^0=Id$ , i am aassuming $O^0=Id $ with $0$ null operator thanks
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Qutient Topologies in Banach Spaces

I hope the title is not misleading. I am currently reading a paper, where the author uses the following argument: First, let $ f:A\rightarrow B $ be a linear and continuous map between Banach spaces $...
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0answers
28 views

The set of all inner derivations $\mathcal{B}^1(A, X)$ is not closed in the set of continuous derivations $\mathcal{Z}^1(A, X)$,

Let $A$ be a complex Banach algebra and $X$ be a Banach $A$-module The set of all continuous $X$-derivations on $A$ is a complex linear space, denoted by $\mathcal{Z}^1(A, X)$. and The set of all ...
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2answers
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Is the limit of the spectral radius the spectral radius of the limit?

Let $A$ be an unital Banach algebra, $x \in A$ and $(x_n)$ a sequence in $A$ converging to $x$. I want to show that $$ \lim\limits_n \rho (x_n) = \rho (x).$$ I can show that $$\limsup \rho(x_n) \leq \...
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0answers
111 views

Does $e^{a+b}=e^{a}e^{b}$ implies that $ab=ba$?

Let $A$ be a Banach algebra we khow that If $ab=ba$ then $e^{a+b}=e^{a}e^{b}$ my question is Does $e^{a+b}=e^{a}e^{b}$ implies that $ab=ba$? Any comment or response is appreciated.
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1answer
64 views

Explaining the group structure in homotopy classes of map $[X,K(H)]$

It is Higon's notes on Index Theory, Ex 3.16, pg 43 where he describes a group structure (where I suppose works in more general context) of the graded $*$-algebra homotopy classes of map $$ [S,K(H)]$...
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1answer
28 views

Simple proof that $x^*x$ is positive in a commutative $C^*$-Algebra

Is there any simple proof (one that does not use continuous functional calculus) for the statement that $\sigma(x^*x) \subseteq [0,\infty)$ for any $x \in \mathcal{A}$ where $\mathcal{A}$ is a ...
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0answers
34 views

some examples of a Banach algebra with given conditions

What are some examples of Banach algerba $A$ satsifying the following two conditions? $1$. $A$ does not have an approximate idebtity. $2$. $A^2=A$. That is, for any $a\in A$, there exist some $b,c\...
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2answers
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Example of a Banach algebra with given property

Please let me know what are some examples of Banach algerba $A$ for which $A^2=A$ ( that is, $a^2=a$ for each $a\in A$). A direct application of the cohen factorization theorem shows that if $A$ has ...
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1answer
75 views

Modern introduction to $C^*$-algebras

Well, I am asking for the references on the subject for those who can't stand the Murphy's book. I have a background in functional analysis (including Banach algebras, functional calculus, Gelfand ...
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7 views

enlarging modules to get innerness of a given derivation

Let $A$ be a Banach algebra, let $E$ be an $A$-bimodule and let $D\colon A\to E\,$ be a continuous derivation. Is it true that by enlarging a bimodule one can make $D$ inner?
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spectrum on finite dimensional Banach algebra

How can I prove that in a finite dimensional complex Banach algebra $A$ we have $\sigma(ab)=\sigma(ba),~\forall a,b\in A$, which $\sigma(a) $ denotes the spectrum of $a$? Every hint is appreciated
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1answer
39 views

Prove that $(\ell^2, \left||.\right\|_2)$ is a commutative and semisimple Banach algebra

Let $(\ell^2, \left||.\right\|_2)$ with the coordinatewise product.Prove that $(\ell^2, \left||.\right\|_2)$ is a commutative and semisimple Banach algebra this is my attempt: Since $(\ell^2, \...
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1answer
55 views

Invertible elements in $l^1(\mathbb Z)$

The vector space $l^1(\mathbb Z)$ with $||x|| = \sum_{n \in \mathbb Z} |x_n|$ and $x * y(t) = \sum_{k \in \mathbb Z} x(k)y(t-k)$ forms a unital complex Banach algebra, with the unit being $\mathbf 1(0)...
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1answer
69 views

Prove that $|f|_X\leq \|f\|$ for every $f\in C(X)$

Let $X$ be a compact space and let $\|.\|$ be an algebra norm on $ C(X)$ Show that $|f|_X \leq \|f\|$ for every $f\in C(X)$ Could anyone please suggest me how to deal with these questions
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1answer
30 views

In Banach algebra $A$ find an example such hat $e^{a+b} \not =e^ae^b$

Let $A$ be a Banach algebra if $ab=ba$ then we have $e^{a+b}=e^ae^b$ without $ab=ba$, I want to find an example such hat $e^{a+b} \not =e^ae^b$ Any help will be greatly appreciated.
2
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1answer
66 views

Bounding || zx - yz || given that || x - y || < M in a Banach algebra.

Let $ X $ be an Banach algebra (not necessarily commutative), and let $ x, y, z \in X $. Suppose that $ \| x - y \| < M $. I want to bound $ \| zx - yz \| $ in terms of $ M $ by writing $ zx - yz ...
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1answer
23 views

A natural Banach algebra where adjoint doesn't preserve the norm?

Are there any natural Banach algebras with some natural operator $A$ where $\Vert A \Vert \ne \Vert A^* \Vert$?
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0answers
69 views

An example of a Banach algebra satisfying given conditions

Is there a non-commutative non-unital Banach algebra $A$ for which $aa_0 -a_{0}a$ lies in the annihilator of $A$ for any $a\in A$? Here $a_0$ is an element of $A$ not belonging to its centre $Z(A)$. ...
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1answer
56 views

Verapoulous Algebra $C(K) \mathbin{\hat\otimes} C(L)$ is a subalgebra of $C(K\times L)$?

Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. ...
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1answer
19 views

Regarding continuous function not in the Disc algebra

Let $D=\{z\in\mathbb{C}: |z|<1\}$. $C(\bar{D})=\{f:\bar D\longrightarrow \mathbb{C}: f \;\text{is continuous on}\; \bar{D}\}$ $A(D)=\{f\in C(\bar{D}): f \;\text{is analytic in} \;D\}$ Can you ...
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0answers
15 views

$C_0(\Delta(l^p))$ description [duplicate]

If $X=l^p$ ,$p \in [1,\infty)$ is nonunitial Banach algebra with coordinate multuplication and $\Delta(l^p)=\{e_n:n \in \mathbb{N}\}$, where $e_n(x)=x_n$ for $x \in l^p$, what is $C_0(\Delta(l^p))$ ?
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Banach algebra $l^p$ is not isomorphic to $C^{*}$ algebra

Consider commutative Banach algebra $l^p$, $p \in [1,\infty)$ with multiplication by coordinates. I know, that $\Delta (l^{p})=\{e_n : n \in \mathbb{N}\}$ - set of canonical functionals. We know that $...
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0answers
13 views

Approximate identity for induced algebra

Let $H$ be a closed subgroup of a locally compact group $G$, and let $A$ be a Banach algebra with an $H$-action $\alpha$. Then we can define an induced algebra $\mathrm{Ind}A$ with a $G$-action by ...
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1answer
41 views

Banach Algebra Isomorphism

Let $X=l^1$ with coordinate multiplication - it is commutative Banach algebra without unit. The Gelfand transformation is defined as $\widehat{x}(e_n)=x_n$ for $x \in l^1$. I would like to prove, that ...
4
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1answer
61 views

Commutative Banach algebras and maximum ideal space

Let $A, B$ be commutative unital Banach algebras and let $\varphi: A \rightarrow B$ be a continuous unital map such that $$\overline{\varphi(A)} = B$$ Let $$\varphi^{*}: \text{Max}(B) \rightarrow \...
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1answer
37 views

When does the spectrum of an element in a Banach algebra with involution lie in the open right half-plane?

Let $A$ be a Banach algebra with involution, $x\in A$ and $t\in {\mathbb R}$ such that $t>\rho(xx^*)$. Show that $\sigma(te-xx^*)$ lies in the open right half-plane. I have no idea! It's obvious ...
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1answer
60 views

Gelfand map is topologically injective

Let $A$ be a commutative Banach algebra and let $$G: A \rightarrow C_{0}(\text{Spec}(A))$$ be a topologically injective map. Recall that the map $T: X \rightarrow Y$ is called topologically ...
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17 views

Lower bound for the speed convergence of the spectrum

Let $A$ be a (complex unital) Banach algebra and $a \in A$ with spectrum $\sigma(a) = \{ 0 \}$ ($a$ is quasinilpotent). For $b \in A$ and $\varepsilon > 0$ consider the linear perturbation $a + \...
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1answer
74 views

Gelfand transformation of $l^p$

I would like to describe Gelfand transofrmation of commutative Banach algebra $l^p(\mathbb{N}),p \in [1,\infty)$ with multiplication define by $(a_n)_n(b_n)_n=(a_n b_n)_n$, but I have no idea, how to ...
2
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1answer
75 views

Spectrum of Banach algebra with coordinate multiplication

Consider $X=l^p$ , $p \in [1, \infty )$. I proved that $X$ with coordinate multiplication is commutative Banach algebra without unit. I have got a problem to find the spectrum of general element of ...
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3answers
78 views

Rudin Functional Analysis, Chater $10$, exercise $12$

Here is the exercise: $12$. Find the spectrum of the operator $T\in {\mathcal B}(\ell^2)$ given by $$T(x_1, x_2, x_3, x_4,...)=(-x_2, x_1, -x_4, x_3,...).$$ I am confused by ${\mathcal B}(\ell^2)$. ...
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0answers
40 views

Lemma 5.1.5 from Garth Dales, Introduction to Banach algebra

The problem is from Garth Dales, Introduction to Banach algebra, chapter 5 and Lemma 5.1.5 Lemma: Let $(A, \|.\|)$ be a unital Banach algebra, let $a\in A$ and let $\epsilon>0.$ then there is a ...