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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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 nuclearity?...
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52 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 $\mathcal{B}^...
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1answer
36 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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2answers
99 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. $\mathcal{L}^1=\mathcal{L}$ , $ \mathcal{L}^2=[\...
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55 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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40 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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46 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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61 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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56 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 $$M_\Lambda(A)=\{(a_{ij})_{i,j\in\Lambda}:\sum_{i,j\in\Lambda}\|a_{ij}\|<...
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52 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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76 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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1answer
40 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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1answer
71 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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1answer
88 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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36 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 +\infty}\|(...
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1answer
77 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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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 $$ \lim_{n\to\infty}\|(x+y)^n\|^{1/n}\le\lim_{n\to\infty}\|x^n\|^{1/n}+\lim_{n\to\infty}\|y^n\|^{1/...
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1answer
74 views

What's the difference between a Banach Algebra and a C*-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 C*-algebras based on my interests. ...
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1answer
74 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 A,\lambda\in\...
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2answers
85 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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198 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
35 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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173 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 ...
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1answer
42 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
24 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
104 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 $\sup_n\|...
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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 ...
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1answer
41 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
101 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
86 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 ...
3
votes
1answer
153 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
28 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 ...
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votes
3answers
56 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 ...
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1answer
20 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 ?
3
votes
2answers
156 views

For a hermitian element $a$ in a $C^*$-algebra, show that $\|a^{2n}\| = \|a\|^{2n}$

Let $\mathcal{A}$ be a $C^*$-algebra. Suppose that $a \in \mathcal{A}$ with the property that $a^* = a$ (that is, suppose that $a$ is hermitian). I would like to show that $\|a^{2n}\| = \|a\|^{2n}...
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1answer
49 views

States: Extension

Definition Define positivity in terms of: $$\omega\geq0:\iff\omega(X^*X)\geq0$$ (This way it serves best for the GNS-construction.) Problem Given a C*-algebra $1\notin\mathcal{A}$ and adjoin a unit ...
2
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1answer
53 views

Norm of rotation matrix as element in $M_2(A)$

Let $A$ be a complex unital Banach algebra and let $R_t=\begin{pmatrix} \cos\frac{\pi t}{2} & -\sin\frac{\pi t}{2} \\ \sin\frac{\pi t}{2} & \cos\frac{\pi t}{2} \end{pmatrix}$. If I consider $...
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1answer
77 views

Nonzero projection in an irredicible C*-algebra of minimal finite rank must have rank one

The following is a part of a theorem in Murphy's C*-algebras and operator theory: Let $A$ be a C*-algebra acting irreducibly on a Hilbert space $H$ and $q$ be a nonzero projection in $A$ of minimal ...
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1answer
44 views

$*-$ isomorphism between two compact spaces $K(H)$ and $K(H')$

The following is a theorem of Murphy's C*-algebras and operator theory: I think we can write the proof more easily than Murphy's. After show that $E'$ is an orthonormal basis for $H'$, define ...
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1answer
35 views

Characterspace of algebra of rational functions on compact subset of $\mathbb C$

Let $K\subset \mathbb C$ be compact and denote with $R$ the closure of rational functions on $K$ w.r.t. $\|\cdot \|_\infty$. Show that the character space of the Banach algebra $R$ is $\{\delta_k:k\...
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1answer
56 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 \...
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1answer
129 views

$p^2=p\in \bar{I}$, I ideal of Banach algebra $\Rightarrow p\in I$

Let $I\subset A$ be a ideal of a Banach algebra $A$. Assume $p\in \overline I$ and $p^2=p$. Show: $p\in I$. Can someone give me a little hint how to solve this, please?
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1answer
43 views

Nonunital C*-Algebras: Closed Image

Given C*-algebras $\mathcal{A}$ and $\mathcal{B}$. Consider a morphism: $\pi:\mathcal{A}\to\mathcal{B}$. Then its image is closed: $\mathrm{im}\pi\subsetneq\overline{\mathrm{im}\pi}$ The proof I ...
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1answer
139 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: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ ...
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1answer
21 views

operators on Hilbert spaces have adjoints

The following is a Theorem of Murphy's C*-algebras and operator theory: In the last line of proof, he claims $u^*$ is linear, but I think it's conjugate linear because for $y_2,x_2\in H_2$, $x_1\in ...
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1answer
40 views

Homeomorphism from $K$ to $\Phi_{C(K)}$

Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that $\...
2
votes
1answer
71 views

If $I$ is a closed ideal in a C*-algebra $A$ and $J$ is a closed ideal in $I$ then $J$ is an ideal of $A$

The following is a remark of Murphy's C*-algebras and operator theory: . I do not know why he uses approximate unit. I think for $a\in A$ and $b\in J^+$, we have $b\in I$ and $b^{1/2}\in I$($I$ is ...
0
votes
1answer
38 views

$\phi(A^+) \subset B^+$ when $\phi: A\to B$ is an isometric linear map

Let $\phi: A\to B$ be an isometric linear map between unital C*-algebras $A$ and $B$ such that $\phi(a^*)=\phi(a)^* (a\in A)$ and $\phi(1)=1$. Show that $\phi(A^+) \subset B^+$. Clearly $A^+ = \{a^*a ...
2
votes
0answers
54 views

Subalgebras of $C_b(X)$ whose elements do not vanish simultaneously at any point

Let $X$ be a completely regular space. How can I find all Banach subalgebras of $C_b(X)$ (all complex-valued bounded continuous functions on $X$) with the property that for every $x\in X$ there ...
2
votes
1answer
61 views

Show that hermitian element $h=\sum p_n/3^n$ generates $ C_0(\Omega)$

Let $\Omega$ be a locally compact Hausdorff space, and suppose that the C*-algebra $C_0(\Omega)$ is generated by a sequence of projections $(p_n)_{n=1}^{\infty}$. Show that the hermitian element $h=\...