29.8 The Gelfand Transform
Definition 29.8.1 (Gelfand Transform).label Let $A$ be a unital Banach algebra, then the Gelfand transform is the homomorphism
Remark 29.8.1.label The Gelfand transform is limited in studying arbitrary Banach algebras, as they may admit no multiplicative functionals. However, these functionals come in abundance in the commutative case.
Proposition 29.8.2.label Let $A$ be a commutative unital Banach algebra and $x \in A$, then:
- (1)
$\Gamma_{A}$ is a contractive homomorphism.
- (2)
$\Gamma_{A}(1) = 1$.
- (3)
$x \in G(A)$ if and only if $\Gamma_{A} x \in G(C(\Omega(A); \complex))$.
- (4)
$(\Gamma_{A}x)(\Omega(A)) = \sigma_{A}(x)$.
- (5)
$\norm{\Gamma_Ax}_{u} = [x]_{sp}$.
Proof, [Theorem 1.1.13, Fol16]. (2): For each $\phi \in \Omega(A)$, $\phi(1) = 1$, so $\Gamma_{A}(1) = 1$.
(3): Since $A$ is commutative, $x \not\in G(A)$ if and only if the ideal generated by $x$ is proper, if and only if there exists a maximal ideal containing $x$, if and only if there exists $\phi \in \Omega(A)$ with $\phi(x) = 0$.
(4): By (1) and (3),
$\square$
Proposition 29.8.3.label Let $A$ be a commutative unital Banach algebra, then the following are equivalent:
- (1)
For each $x \in A$, $\normn{x^2}_{A} = \norm{x}_{A}^{2}$.
- (2)
$\Gamma_{A}$ is an isometry.
Proof. (1) $\Rightarrow$ (2): For each $x \in A$, by the spectral radius formula and (5) of Proposition 29.8.2,
(2) $\Rightarrow$ (1): For each $x \in A$, by (5) of Proposition 29.8.2,
$\square$
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