Questions tagged [gelfand-representation]
Gelfand representation is a way of representing commutative Banach algebras as algebras of continuous functions.
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Trying to understand the functional calculus of a unitary operator.
I am trying to understand the theorem on this Wikipedia article.
It is stated as follows.
Theorem. Let $x$ be a normal element of a $C^*$-algebra A with an identity element e.
Let $C$ be the $C^*$-...
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Proof of $C(X)\otimes_{\text{min}}C(Y)\cong C(X\times Y)$ via Gelfand spectrum
i want to understand a proof of $C(X)\otimes_{\text{min}}C(Y)\cong C(X\times Y)$ for compact Hausdorff spaces $X,Y$ via the Gelfand transform. Using the latter, it suffices to show that $\Omega(C(X)\...
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Little Gelfand-Naimark theorem
Let $\mathcal{A}$ be a unital commutative C* algebra. Its Gelfand transform
$$ \hat{a}: \mathcal{A} \rightarrow C(\hat{\mathcal{A}}) $$ is an isometric *-isomorphism.
(partially because its range is ...
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How to show $\phi $ is an injective map?
I'm trying to show that $\phi$ is an injective map for this i'm trying to show $\phi(t_1)=\phi(t_2)\implies t_1=t_2$
let $$\phi(t_1)=\phi(t_2)$$ $$\implies \varphi_{t_1}=\varphi_{t_2}$$ $$\implies \...
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Fourier / Gelfand transform vanishes at infinity?
I've come across the fact that the Fourier transform (or, more general, the Gelfand transform) vanishes at $\infty$. See for example "Principles of Harmonic Analysis" by Deitmar and ...
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How to show that R is a closed ideal in B?
Let $B$ be a commutative Banach algebra and $R=\left\{ f\in B:1+\lambda f\text{ is invertible for any }\lambda \in C\right\}$.
Show that R is a closed ideal in B
I used Gelfand representation to ...
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Character Space of $C^1[0,1]$ and Gelfand Representation
I have recently been working through some exercises in Murphy's "C*-Algebras and Operator Theory," and I am having some trouble with Exercise 10 in Chapter 1. The exercise is as follows:
Let ...
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An example which disproves $\|p(T)\|_{op} = \|p\|_\sup$ in arbitrary Banach spaces (Gelfand theory).
Let me go straight to the question and then write some discussion about it.
Question: Does there exist a Banach space $X$, an operator $T:X\rightarrow X$ with operator norm $\|T\|_{op}\leq 1$ and a ...
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Let $G=SO(3)$ and $\theta:G\to G$ defined by $\theta(g)=JgJ$ where $J=\text{diag}(-1,1,1)$. Prove that, $\theta(g)\in Kg^{-1}K$ where $K=SO(2)$
Here $K=\left\{\begin{pmatrix}1 & 0\\ 0 & T_1\end{pmatrix}:\ T_1\in SO(2)\right\}$. $K$ is a compact subgroup of $G$. The above problem is part of the proof of $(G,K) $ is Gelfand pair.
I've ...
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Direct sum of direct integrals
Let $(X,\mu)$ be a measure space, and $(H_x)_{x \in X}$ is a "field" of Hilbert spaces, one can form the direct integral $\int^{\oplus} H_x d\mu(x)$ which is a Hilbert space as well.
When $(...
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How to prove this expression is a polynomial
How can we prove that the expression
$$\gamma_{m k}(\lambda)=\sum_{i=1}^{m}\left(\lambda_{m i}+m-1\right)^{k} \prod_{j \neq i}\left(1-\frac{1}{\lambda_{m i}-\lambda_{m j}}\right)$$
is a polynomial in ...
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Proof of Corona problem for $\mathbb{D}$
Denote by $H^\infty$ all analytic and bounded functions on the unit disk $\mathbb{D}:=\{x\in\mathbb{C}:|x|<1\}$. I want to show that $\Delta_0:= \{[f\to f(w)]:w\in\mathbb{D}\}\subseteq\Delta(H^\...
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How to describe the Gelfand transform of the Banach algebra of complex Borel measures on the real line?
Let $M$ be the Banach algebra of all complex Borel measures on $\mathbb{R}$. To be clear,
Norm: $\| \mu \| = |\mu|(\mathbb{R})$, where $|\mu|(E)$ is the total variance.
Product: $(\mu \ast \lambda)(E)...
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Gelfand-Kolmogorov-type theorem for $C_0(X)$?
I am looking for good references for Gelfand-Kolmogorov-type theorems for different function spaces—the space of vanishing functions, in particular. Explicitly, I am after a proof of the following ...
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What is the cardinality of $K$, where $L_\infty[0,1]=C(K)$?
By Gelfand representation of (real) C$^*$-algebras, it is known that $L_\infty[0,1]$ is isometrically isomorphic to $C(K)$, for some compact Hausdorff $K$. By looking at the proof, $K$ is actually ...
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Gelfand-Naimark theorem fails for Banach algebras?
The Gelfand-Naimark theorem says that if $A$ is a commutative unital $C^*$-algebra, then $C(Spec(A))=A$, where $Spec(A)$ is the set of all characters on $A$.
Does the theorem fail for commutative ...
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Gelfand space of a commutative unital Banach algebra is weak* compact
Before I come to my actual questions, I want to give some context (e.g. definitions, ...).
(1) Let $X$ be a normed space over a field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$. Then we define $ X' :=...
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Learning roadmap and prerequisites for Isbell duality
I'm looking for a roadmap to learning about Isbell duality. I know a reasonable amount about several of the "specific" dualities (Gelfand duality, AffSch - CRing, frames - locales, etc), ...
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When exactly is the character space of a Banach algebra empty?
It is well-known that the character space, (i.e. the set of multiplicative characters) of a commutative, unital Banach algebra is non-empty.
But is there a complete characterization of when exactly ...
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Why is the spectrum of $L^\infty[0,1]$ inseparable?
In the Gelfand correspondance any measurable subset of $[0,1]$ corresponds to a clopen subset of $\text{Spec}(L^\infty[0,1])$, so clearly the space contains a bunch of clopens. I'm not sure if this is ...
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Must an abstract $C^*$-algebra that is an integral domain be a field?
I'd like to understand a little bit better why the maximal (as opposed to prime) spectrum is the appropriate notion of spectrum for the theory of $C^*$-algebras.
The canonical answer is that $C^*$-...
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How exactly is the Fourier transform the same as the Gelfand transform?
This answer states the Fourier transform is the Gelfand transform on the Banach algebra $L^1(G)$ with convolution. I've read the resource linked in the answer, but I still have some confusion.
My ...
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Spectrum of unital, commutative C star algebra
According to the Wikipedia article on the Gelfand Represenetation (C* algebra section), the spectrum of a commutative C* algebra $A$ (the non-zero *homomorphisms $\phi : A \rightarrow \mathbb{C}$)
i)...
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Relationship between weak topology and Gelfand topology (Banach space theory)
I don't have much background in functional analysis, so wanted to check whether my thinking about the weak topology and Gelfand topology is correct when it comes to C* algebras/Banach spaces.
My ...
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An outline of proving Gelfand-Neimark Theorem for Commutative $C^{*}$ algebras
I would like to look at a sketch or an outline of the proof of the Gelfand-Neimark Theorem for commutative $C^{*}$ algebra's. I am doing a final year essay and I just want the major points or ideas to ...
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Banach subalgebras and * subalgebras of a C* algebra
Given a unital C* algebra $A$, or more specifically, $A:=\mathcal L(H)$ be the bounded linear operators of a certain Hilbert space $H$.
How do its subalgebras look like if one only considers the pure ...
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Gelfand representation and extremally disconnected spectrum?
Let $S$ be a set, $\Sigma$ an algebra on $S$ and consider the Banach space $B(S, \Sigma)$ of bounded $\Sigma$-measurable functions $f : S \to \mathbb{R}$. Denote by $K_\Sigma$ the Gelfand spectrum of $...
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Question about abelian C*-algebras and the corresponding Gelfand transform: $C_{0}(\Sigma)$ or $C(\Sigma)$.
In Murphy, Theorem 2.1.10, they say that any abelian non-zero C*-algebra $A$ is $\ast$-isomorphic to $C_{0}(\Sigma_{A})$, where $\Sigma_{A}$ is a maximal ideal space.
In Conway, Corollary VIII.2.2, ...
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When can every element of a Hilbert space be represented in terms of the spectral resolution of a self-adjoint operator?
Let $(L, D(L))$ be a self-adjoint operator in a Hilbert space $\mathfrak{H}$ (in particular, I am assuming this operator is the generator of $C_0$-contraction semigroup on $\mathfrak{H}$), and let $E$ ...
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Gelfand transform and unitization misunderstanding?
This is my first post on StackExchange, so please let me know if I should modify my question in any way, or if this is not a reasonable question to ask here. I have tried to find other questions ...
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From a Poisson algebra to its geometric dual
I am trying to recover a Poisson manifold from its algebraic (commutative) dual. So I looked at the side of Gelfand transformation (since a Poisson algebra is a commutative Banach algebra). Gelfand ...
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Functional calculus of several variables
It is well know that for a normal element $a$ of C*-algebra $A$ there exists functional calculus namely there is a *-homomorphism $C(\sigma({a})) \to A$ uniquely determined by sending $z \mapsto a$.
...
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What is the purpose of making no reference to operators on a Hilbert space (Gelfand-Naimark)?
I read from here pag. 209
The residue ring of an *-ring is an *-ring itself. Hence follows that if R is a closed *-subring of the ring of operators in Hilbert space, then any its residue ring can ...
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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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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 ...
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What is the Gelfand-Naimark representation of functions that don't vanish at infinity?
The Gelfand-Naimark theorem says that every commutative C*-algebra is isometrically isomorphic to $C_0(X)$, the set of continuous functions $f:X\rightarrow\mathbb{C}$ that vanish at infinity, for some ...
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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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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 ...
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Identification of Gelfand spectrum, $\Sigma(\mathcal{A})$, with $\sigma(A)$, spectrum of Banach Algebra element
I am reading Mathematical Structure of Quantum Mechanics A short Course for Mathematicians by F. Strocchi.
My question is about the text after Proposition 1.5.3. The proposition is that
For a ...
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Gelfand-Naimark theorem and compactifications
Let $X$ be a Hausdorff space and let $C(X)$ denote the set of all continuous and bounded functions from $X$ to $\mathbb{C}$. It is a well-known fact that $C(X)$ forms a unital and abelian $C^*$-...
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Gelfand spectrum of a Banach algebra generated by a single element
Let $A$ be a Banach algebra generated by $a$ which means it is a closure of $span \{1, a, a^2, ... \}$. The problem is to show that its Gelfand spectrum $\Omega(A)$ is homeomorphic to the spectrum of $...
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Unital homomorphism to semisimple Banach algebra is automatically continuous
I need to prove that any unital homomorphism $\phi: A \to B$, where $A$ is unital Banach algebra and $B$ is semisimple Banach algebra is continuous.
The definition of "semisimple" I know is that ...
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The induced representation of a trivial representation
How can I show that the induced representation of a trivial representation of a subgroup is the permutation representation on its cosets?
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Range of the Gelfand transform on a non-unital Banach algebra
Let $\mathcal{A}$ be a non-unital commutative Banach algebra. Consider the Gelfand transform
\begin{align*}\Gamma_{\mathcal{A}}:\mathcal{A}&\to C(\sigma(\mathcal{A}))\\
x&\mapsto \hat{x}
\end{...
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Character space of $C^{0}(X)$
Let X be a locally compact Hausdorff topological space.
Then I need to find the character space of $C^{0}(X)$. A reference will be very helpful
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Spectral Permanance in $C^{*}$ algebras
Problem:
Let $A$ be a unital $C^{*}$ algebra and let $B$ be sub $C^{*}$ algebra of A with possibly a different unit. Then I need to prove that for any $b$ $\in$ $B$
spectrum($b,B$)$\cup$ {$0$} $\...
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If Banach Algebras are isomorphic, then their corresponding maximal ideal spaces are homeomorphic
Let $A$ ans $B$ be two unital commutative Banach algebras which are isomorphic to each other.
Then I need to prove that their maximal ideal spaces are homeomorphic,
where by maximal ideal space of $A$,...
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Trying to understand how the Gelfand spectrum relates to the usual notion of spectrum of an operator?
I am trying to understand how the Gelfand spectrum relates to the usual notion of spectrum of an operator. I have tried to construct a simple artificial example to help with understanding but I am ...
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A question on the proof of Gelfand-Kazhdan theorem.
Consider $F=\Bbb Q_p$ and $G=GL(n, F)$ and $\theta \in \operatorname{Aut(G)}$ s.t $\theta(g)=(g^t)^{-1}$
Now Gelfand-Kazhdan theorem says that:
For any representation $(\pi, V)$ of $G$, if we ...
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Can we characterise $X$ being separable in terms of $C(X, \mathbb R)$?
Consider the real line $\mathbb R$ and the Banach algebra of bounded real functions $C_b(X, \mathbb R)$.
By Gelfand duality the space of maximal ideals of $C_b(X, \mathbb R)$ becomes a compact ...
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