Questions tagged [gelfand-representation]

Gelfand representation is a way of representing commutative Banach algebras as algebras of continuous functions.

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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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1answer
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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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Gelfand representation of $C_0(X)$

In the book "A course in commutative Banach Algebras," Kaniuth shows that the Gelfand representation of $C_0(X),$ where $X$ is locally compact is the identity map. The argument is as follows: The ...
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Could you help me show the corresponse in details?Thanks pretty much in advance!

Let $\mathcal{A}:=\{f:\mathbb{Z}\rightarrow C:\|f\|:=\sum_{n\in Z}|f(n)|2^{|n|}<\infty\}$ under the usual function addition and scalar multiplication .Then with function multiplication : $f\ast g(...
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Characteristic function on a certain clopen subset vanishes at infinity

Let $A$ be a C*-algebra, $B\subseteq A$ an abelian C*-subalgebra, $\alpha \in Aut(A)$ and $X:=Spec(B)$. Let $x\in A$ with $xb=\alpha\left(b\right)x$ for all $b\in B$. Let $f \in C_0(X)$ the positive ...
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Approximate unit on $C_0(X)$ converges uniformly on compact sets

Let $(e_{\lambda})_{\lambda \in \Lambda} \subseteq C_0(X)$ with $X$ locally compact an (increasing) approximate unit. I assume that for every compact $K\subseteq X$ we have $\left\Vert 1-e_{\lambda}\...
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1answer
138 views

Inequality Concerning Positive Elements in a C$^{*}$-algebra of Norm at Most One

I am studying the GNS construction and in part of a proof in one of the theorems along the way (theorem 3.3.3. in Murphy's book), the following statement is made: Now suppose that $a$ is positive ...
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What space corresponds to the localisation of the ring of continuous functions?

Suppose $A$ is a commutative Banach algebra. By Gelfand duality there is a compactum $X$ such that $A = C(X)$ is the ring of continuous functions. The space $X$ can be recovered as the space of ...
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If $a$ is normal in a $C^*$-algebra $A$, then $\tau(a^*)=\overline{\tau(a)},\forall \tau\in\Omega(A)$?

I'm reading Murphy's text. I don't understand that why the underlined map preserves adjoints. ($\Omega(A)$ is the character space of $A$.) Can someone help me? Thanks a lot. (Is it because $f(a)$ is ...
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68 views

Proving the existence of a Gelfand transform

If $A$ is a commutative Banach algebra and $f$ is holomorphic on $\Omega\subseteq\mathbb{C}$, where $\Omega$ contains $\mathscr{R}(\hat{x})$, I want to prove the existence of a $y\in A$ with $\hat{y}=...
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Using regularity of a Banach algebra

The following comes from Katznelson's An Introduction to Harmonica Analysis Chapter VIII, pages 236 to 239. Definition : A function algebra $B$ on a compact Hausdorff space $X$ is regular if, given ...
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Is the range of the Gelfand transform closed?

Let $A$ be a commutative unital Banach algebra. Consider the Gelfand map $\Gamma:A\longrightarrow C(M_A)$, $\Gamma(a)=\hat{a}$, where $M_A$ is the Character space of $A$. Is the image of the Gelfand ...
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Can you give an examples of non commutative non C*algebras?

Are there examples apart from $B(X)$ where $X$ is not a Hilbert space and not finite dimensional. Do they have a characterization or representation?
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153 views

In Larsen's proof of a theorem regarding $L^\infty(X, S, \mu)$

I was reading Larsen's proof that if $(X,S,\mu)$ is a positive measure space then $L^\infty(X,S,\mu)$ is isometrically isomorphic to $\mathcal{C}(\Delta(L^\infty(X,S,\mu)))$ and $\Delta(L^\infty(X,S,\...
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Gelfand transform on functions

The Gelfand transformation identifies function spaces $C_0(X)$ for locally compact Haussdorff $X$ with commutative $C^*$ Algebras. Additionally there is a statement that if $f: X \to Y$ is a proper ...
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Where does Gelfand Theory fail for non-commutative algebras.

I'm trying to get my head around Gelfand theory, and I can't seem to find the subtleties between commutative and non-commutative algebras. Why is there not a one-to-one correspondence between maximal ...
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141 views

Gelfand transform of $a \in A$ vanishes at infinity?

Let $A$ be a non-unital commutative Banach algebra defined over the field of complex numbers. Given $a \in A$, one defines the function $\widehat{a}:\Phi_A\to{\mathbb C}$ by $\widehat{a}(\varphi)=\...