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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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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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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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 ...
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?