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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Tensor norm for matrix algebra over an $L^p$ operator algebra

This is a question about whether a certain tensor norm has a certain property. The setting is that of $L^p$ operator algebras (i.e. norm-closed subalgebras of $L^p(X,\mu)$ for some measure space ...
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2 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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18 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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12 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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172 views

Tensor product of algebra

Can we find define a norm on tensor product $C(X) \otimes C(Y)$ such that the norm completion of $C(X)\otimes C(Y)=C(X\times Y)$ And can we define a norm on tensor product $L^1(X)\otimes L^1(Y)$ such ...
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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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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 ...
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1answer
26 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 ...
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1answer
288 views

A semisimple commutative Banach algebra with a non-semisimple quotient

I am looking for an example of a semisimple commutative Banach algebra which admits a non-semisimple quotient. Attempt from the comments: "I take $A$ to be the algebra of all continuously ...
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2answers
53 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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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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28 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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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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23 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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1answer
53 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 ...
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46 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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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 ...
2
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2answers
35 views

Bounded approximate identities in one-sided ideals of $L_1(G)$

Let $G$ be a locally compact group and consider its $L_1$-group algebra, that is, $L_1(G)$. It is easily seen that $L_1(G)$ has a two-sided bounded approximate identity: just use a basis of compact ...
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100 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 ...
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47 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 ...
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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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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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58 views

Semigroups: Nonexample

Given a C*-algebra $\mathcal{A}$. Consider a *-derivation $\delta$. Does it always generate a group: $$\tau(t)=e^{it\delta}$$ But a group of automorphisms is a contraction group: ...
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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
32 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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1answer
24 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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20 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
357 views

The disk algebra and continuous functions

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 ...
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30 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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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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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 $$ ...
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1answer
38 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. ...
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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 ...
3
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2answers
119 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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8 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 ...
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1answer
19 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 ($V, ||$ $ || $) ...
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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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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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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 ...
12
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1answer
530 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
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1answer
77 views

Convergence Radius: Non-Analyticity

Why is a function certainly nonanalytic at some point on the radius of convergence? I mean considering a power series around somewhere and if theres a power series expansion at every point on circle ...
2
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2answers
145 views

A closed ideal in a commutative Banach algebra $C(X)$

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 (necessarily ...
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1answer
50 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 ...
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2answers
75 views

(Commutative Banach algebra) Prove that $G(\mathcal A)$ be an open set in $\mathcal A$.

UPDATE I have a problem: Let $\mathcal A$ be a commutative Banach algebra. Denote $G(\mathcal A)$ is the set of all invertible elements in $\mathcal A$. Prove the following assertions: ...
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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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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
46 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 ...