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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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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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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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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Vector States and GNS Construction

Let me reword my question. Why is it that when $A$ is a $C^*$ normed algebra and $\mu$ is a state, then $\mu(a)=\langle L_a\xi,\xi\rangle_\mu$ for some $\xi$. Now I know that there is an approximate ...
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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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28 views

Matrix norm on Banach algebra

Let $A$ be a unital Banach algebra. Consider the matrix 2-norm on $M_n(A)$, i.e. for a matrix $T$, $\|T\|=\sup_{x\neq 0}\frac{\|Tx\|_2}{\|x\|_2}$ where ...
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1answer
27 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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29 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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30 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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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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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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32 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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30 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 ...
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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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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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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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25 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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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
17 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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12 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
34 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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61 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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36 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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32 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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39 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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5 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 $\lambda-a'$ ...
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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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15 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 ...
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31 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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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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44 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
17 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 ?
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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}\| = ...
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37 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 ...
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1answer
41 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
61 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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34 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
32 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 ...
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1answer
35 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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53 views

Does the equality $(T^{\ast})^{\alpha}=(T^{\alpha})^{\ast}$ hold?

Let $H$ be a complex Hilbert space and $B(H)$ be all bounded linear operator on $H$. Let $T\in B(H)$. Can I say $(T^{\ast})^{\alpha}=(T^{\alpha})^{\ast}$ for $\alpha\geq1$ where $\ast$ means ...
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$p^2=p$ in closure of ideal $I$ of Banach algebra implies $p\in I$.

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
38 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
105 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
19 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
32 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 ...
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1answer
37 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 ...
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1answer
20 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 ...
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50 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 ...
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1answer
24 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 ...
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1answer
43 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...