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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41 views

Can $ {L^{1}}(G) $ be a $ C^{*} $-algebra?

Let $ G $ be a locally compact abelian group. Then $ {L^{1}}(G) $ is a commutative algebra when equipped with convolution. Is there an involution $ ^{*} $ on $ {L^{1}}(G) $ so that it becomes a $ ...
4
votes
1answer
18 views

A property of a ideal of Banach algebras

Let $B$ be a Banach algebra and $A$ be a bi-ideal of it. Suppose that for any $b\in B$, $Ab=\{0\}$ implies $b=0$. Now could we say that for some $c\in B$ if $cA=\{0\}$ then $c=0?$
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1answer
24 views

Banach algebra norms on $M_n(A)$

Let $A$ be a Banach algebra (not sure whether I need $A$ to be unital). I saw the claim that all Banach algebra norms on $M_n(A)$ with continuous projections on entries are equivalent. How does one ...
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0answers
10 views

When do injective and projective tensor norms agree?

For $C^*$-algebra tensor products, one talks about the min and max tensor norms, and they agree when one of the $C^*$-algebras is nuclear. For general Banach algebras, what is the analog of ...
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0answers
12 views

Operator norms and tensor norms on $M_n(A)$

If $A$ is a (unital) complex Banach algebra, then $M_n(A)$ can be equipped with the various operator norms (with respect to $p$-norms, say for $1<p<\infty$) and these are equivalent Banach ...
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27 views

Weak topology and the closed unit ball

I want to prove that there is no neighbourhood of $0$ in the closed unit ball. I can use pointwise and Banach-Alaoglu theorems to prove it.
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18 views

Determining primitive ideal space of C* algebra

What is the general way of determining the space of primitive ideals of the C* algebra if there is any? Thanks.
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27 views

Continuous homorphism from $(\mathbf{R},+)$ to group of invertible elements in Banach algebra is differentiable

Let $A$ be a Banach algebra with $1$ and $\varphi\colon \mathbf{R}\to A$ be continuous such that $\varphi(0)=1$ and $\varphi(x+y)=\varphi(x)\varphi(y)$ for each $x,y\in \mathbf{R}$. The claim is ...
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0answers
21 views

if $f$ is in Banach space, then $\nabla f $ is in the dual space?

I am not very deep in advanced real analysis. Could you help me decipher the following two phrases hold? 1) if $f$ is in Banach space $\mathcal{B}$, then $\nabla f $ is in the dual space ...
3
votes
1answer
28 views

Multiplicative linear functionals on subalgebras

If $A$ is a commutative $C^\ast$ algebra and $C$ is a $C^\ast$ sub algebra of $A$ is it true that the characters on $C$ are just restrictions of characters on $A$. The reason I am asking this because ...
2
votes
2answers
55 views

Algebra is Generated by nilpotent Lie Algebra

$X$ Banach space. In $B(X)$, we can define a Lie product $[ , ]:[T_1,T_2]=T_1T_2-T_2T_1$ for any $T_1,T_2 \in B(X).$ Let $\mathcal{L}$ Lie Algebra. $\mathcal{L}^1=\mathcal{L}$ , $ ...
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2answers
29 views

realizing/ understanding $C^*(\phi_g(C([0,1])))$ and "support projection of an element of a $C^*$-algebra

someone asked me the following question but I don't know the answer, but and I'm interested in it too. Let $C([0,1])=\{f:[0,1]\to\mathbb{C}; \text{f is continuous}\}$, $g\in C([0,1])$ be a positive ...
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0answers
24 views

Banach algebras for which the Gelfand transform is 1-1 but not onto

Are there examples of Banach Algebra's for which the Gelfand transform is 1-1, (ie: the intersection of all maximal ideals of the algebra is $\{0\}$) but not onto? Context Page 96 of Kaniuth's ...
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1answer
35 views

almost unital Banach algebra's

Let $A$ be an "almost unital Banach algebra", in the sense that it satisfies all the usual axioms but not necessarily that $\|1\|=1$. From the product inequality $\forall x,y \in A$ \begin{equation} ...
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0answers
24 views

making a dense set of bump function

Can we write a dense set of bump function by continuous functions vanishing at 0? Define $$\begin{align*} \varphi &:L^2[0,1] \to L^2[0,1], \\ &(\varphi f)(x)=‎\int_0^x f(t)dt. ...
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vote
1answer
54 views

preserving problem

Define: $$\begin{align*} \varphi &:L^2[0,1] \to L^2[0,1], \\ &(\varphi f)(x)=‎\int_0^x f(t)dt. \end{align*}‎$$ Is it true that, if ‎‎$‎‎B$ is a dense ‎subset ‎of ‎‎$‎‎L^2[0,1]$, then ...
3
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0answers
47 views

Ideal in Matrix algebra

Let A be a Banach algebra and suppose that $\Lambda$ be a nonempty set. With the following product and norm the matrix space ...
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1answer
48 views

Quotient Rings of Algebras

So let us take the commutative Banach algebra $B=\mathscr{l}^1(\mathbb{Z}_n)$ over $\mathbb{R}$ with convolution as multiplication $(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ...
0
votes
1answer
59 views

Topological modules and relative homological algebra.

This question might be a bit dumb but I'm tired right now and this is just going over my head at the moment, in "The homology of Banach and topological algebras" Helemskii said that relative ...
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votes
1answer
35 views

If $X$ is compact Hausdorff, is every isomorphicm $\mathcal{C}(X) \to \mathcal{C}(X)$ continuous?

If $X$ is a compact Hausdorff space, is then every isomorphism from ${\mathcal C}(X)$ onto ${\mathcal C}(X)$ is continuous?
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0answers
16 views

Lifting an element isometrically (or contractively) from a quotient of a Banach algebra

Let $A$ be a (unital) Banach algebra and let $J$ be an ideal. Let $A_r$ be a closed linear subspace of $A$. Suppose that the continuous linear bijection $A_r/(J\cap A_r)\rightarrow (A_r+J)/J$ is an ...
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0answers
13 views

Pullback of the norm on the holomorph by the Riesz functional calculus

Conway states that the holomorph $H(a)$ of an element $a$ of a Banach algebra is not a Banach algebra. Let $||f||=||f(a)||$ for any $f\in H(a)$. We need to see that this "norm" is separates the ...
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1answer
25 views

GNS Construction on non-unital algebra

STATEMENT: If A has a multiplicative identity 1, then it is immediate that the equivalence class $ξ$ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation. If $A$ is ...
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21 views

Inductive/Projective Limits of Topological Algebras

It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance, For $k \ge 0$ and $K_n$ compact ...
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1answer
34 views

Quaternions as a counterexample to the Gelfand–Mazur theorem

It seems that by the Gelfand–Mazur theorem quaternions are isomorphic to complex numbers. That is clearly wrong. So where is the catch? I think that I found the problem but it seams so subtle that ...
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vote
0answers
31 views

Equivalent statements involving 'little o'

Let $A$ be a Banach algebra and $a\in A$. $\|z(z-a)^{-1}\|=1+o(\frac{1}{z})$ as $z\rightarrow +\infty$ iff $\|(z-a)^{-1}\|^{-1}=z+o(1)$ as $z\rightarrow +\infty$. i.e., $lim_{z\rightarrow ...
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1answer
40 views

Are all function transforms special cases of Gelfand's transform?

Reading about Gelfand-Naimark theorem I've seen that the Fourier transform is a special case of Gelfand transform for the space $L^1(\mathbb{R})$ with the convolution product. In a related question on ...
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0answers
7 views

Banach space X and a closed subspace V equipped with the same norm.

In the first one, you have a Banach space X and a closed subspace V equipped with the same norm. Then, one knows that V ∗ ≅X ∗ /V ⊥ (that is, take the quotient space w.r.t. the annihilator of V ). ...
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0answers
24 views

an inequality in Banach algebra [duplicate]

Let $(V, \| \ \|)$ be a Banach algebra. Given two elements $x,y\in V$ satisfying $xy=yx$, prove that $$ ...
1
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1answer
39 views

What's the difference between a Banach Algebra and a CStar Algebra?

I'm currently looking at going into a PhD program in mathematics and need to decide on a specialization. In meeting with my advisor he pushed me into looking at Cstar algebra's based on my interests. ...
2
votes
1answer
39 views

Norms on unitization of nonunital Banach algebra

Let $A$ be a nonunital Banach algebra and denote by $A^+$ the unitization of $A$. One commonly used Banach algebra norm on $A^+$ is given by $||(a,\lambda)||=||a||+|\lambda|$ (where $a\in ...
3
votes
2answers
56 views

Approximate Identities

Statement : Let $U$ be a C* algebra and$\lambda=\left\{A\in U:A\geq 0, ||A||<1\right\}$. If $B\in \lambda$ then if $X\in U$ $$||X^*(I-B)^2X||\leq ||X^*(I-B)X||$$ For reference this is from ...
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0answers
9 views

Automorphisms of the Banach algebra of continuous functions of bounded variation on [0,1]

Let CV be the Banach algebra named in the title with pointwise multiplication. If \varphi is a homeomorphism of [0,1] such that \varphi is an invertible element of C^1[0,1] then f \to f\circ \varphi ...
3
votes
2answers
120 views

Maximal Ideal Spaces

STATEMENT: Let $C_b(\mathbb{R})$ be the $C^*$-algebra of bounded continuous functions on $\mathbb{R}$. Let $A$ be the $C^*$-subalgebra of $C_b(\mathbb{R})$ generated by $C_∞(\mathbb{R})$ together with ...
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1answer
28 views

Banach algebra of homomorphisms

Let $E,F$ be Banach spaces. Is it always true that $\mathrm{Hom}(E,F)$ is Banach algebra ?
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2answers
49 views

A question about locally compact Hausdorff space

If $X$ is a locally compact Hausdorff space, $C_{0}(X)$ denotes the set of continuous functions from $X$ to $\mathbb{C}$ vanishes at infinity. This is a basic example in C*algebra. My question is Why ...
0
votes
1answer
21 views

Specific question on Banach space over nonarchimedean field

Let K be an nonarchimedean field. We let $Ban(K)$ denote the category of $K$-Banach space with continuous linear maps and let $C$ be the category of normed K-Banach spaces ($V, ||$ $ || $) ...
0
votes
1answer
17 views

Let $A=\mathbb{C}$ and $E=F=C[0,1]$. Are $E=F$ Hilbert $\mathbb{C}^*$-modules?

Let $A=\mathbb{C}$ and $E=F=C[0,1]$. Are $E=F$ Hilbert $\mathbb{C}^*$-modules? clearly $E=F$ are $C^*$-algebra
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1answer
53 views

Maximal ideal space

Let $X$ be a compact space, $x_0\in X$, and define $$A=\{\{f_n\} ; f_n\in C(X), \sup_n\|f_n\|<\infty, and \{f_n(x_0)\} \text{ is a convergent sequence} \} $$ If $\|\{f_n\}\|$ is defined as ...
0
votes
0answers
68 views

Is this true? If ‎$‎\varphi\in ‎\mathcal{B}_{‎\mathcal{A}}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection

Is this true? If ‎$‎\varphi\in ‎\mathcal{B}_{‎\mathcal{A}}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection, why?. Such that $E$ and $F$ are right Hilbert ‎$‎‎‎\mathcal{A}‎$‎-modules ...
1
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1answer
40 views

Is this true? If ‎$‎\varphi\in ‎\mathcal{B}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection. Why?

Is this true? If ‎$‎\varphi\in ‎\mathcal{B}(E,F)‎$‎ and preserving density, then $‎\varphi‎$‎ is surjection, such that $E$ and $F$ are Banach space and $\mathcal{B}(E,F)‎$‎ is the set of all bounded ...
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vote
0answers
39 views

For a Banach algebra $\mathcal{A}$ and an idempotent $e$, show that $\mathcal{A}e$ is a Banach algebra

I am studying the spectral theory of operators on Banach and Hilbert spaces, reading through Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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1answer
47 views

Understanding a proof from Conway: showing existence of idempotents using functional calculus

I am studying the spectral theory of operators on Banach and Hilbert spaces, making use of Conway's A Course in Functional Analysis. In section VII.4, Conway states Proposition 4.11 as a consequence ...
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votes
0answers
7 views

Norm of the resolvent in the quotient algebra

Let $A$ be a Banach algebra with identity and $J$ be a two-sided ideal of $A$. It is known that $\sigma(a')\subseteq\sigma(a) \, \forall a\in A$, where $a'$ is the coset $a+J$. Suppose $\lambda-a'$ ...
0
votes
0answers
15 views

Banach Algebra convergence

I am taking a courses on Differential Calculus and I would like to know the details in the steps of the proof (1) & (2). Also I would like some details on how to solve the exercise below. (3) ...
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0answers
22 views

Spectrum of a C* subalgebra generated by a normal element

Let $A$ be a unital C*-algebra and $a$ be non-invertible normal element in $A$. Suppose $B=C^*(a)$, i.e., the commutative C*-subalgebra that is generated by $a$. Will ...
3
votes
1answer
46 views

Compact operators and essential spectral radius

Let $E$ be an infinite-dimensional complex Banach space. Let $\mathcal{L} (E)$ be the space of endomorphisms of $E$, endowed with the operator norm. Then $\mathcal{L} (E)$ is a unital Banach algebra ...
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0answers
23 views

isometrically isomorphism [duplicate]

How can embed separable Banach to Cb(X)(family of all bounded continuous functions on topological space X) with non metrizable X ? If X is locally compact or Tychonof is very well. note : We know ...
2
votes
3answers
49 views

Semi-direct decompositions of Banach algebras

Let $A$ be a Banach algebra and I an ideal of it. Is there always a subalgebra $B$ of $A$ such that A can be written as $A=B\oplus I$ where $\oplus$ denotes direct sum? If not, in what conditions we ...
1
vote
1answer
18 views

Diffeomorphism in Banach algebra

Let $X$ be an algebra $C([0,1],\mathbb{R})$. Let define $F:X\ni f \mapsto f(0)f \in X$ What is the biggest $r$ such that $F$ is $C^r$-diffeomorphism ?