# 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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### 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