37.8 The Gelfand Transform
Definition 37.8.1 (Gelfand Transform).label Let $A$ be a unital Banach algebra, then the Gelfand transform is the contractive homomorphism
Remark 37.8.1.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 37.8.2.label Let $A$ be a commutative unital Banach algebra and $x \in A$, then:
- (1)
$\Gamma_{A}(1) = 1$.
- (2)
$x \in G(A)$ if and only if $\Gamma_{A} x \in G(C(\Omega(A); \complex))$.
- (3)
$(\Gamma_{A}x)(\Omega(A)) = \sigma_{A}(x)$.
- (4)
$\norm{\Gamma_Ax}_{u} = [x]_{sp}$.
Proof, [Theorem 1.1.13, Fol16]. (1): For each $\phi \in \Omega(A)$, $\phi(1) = 1$, so $\Gamma_{A}(1) = 1$.
(2): 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 $\phi(x) = 0$.
(3): By (2),
$\square$
Proposition 37.8.3.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 (4) of Proposition 37.8.2,
(2) $\Rightarrow$ (1): For each $x \in A$, by (4) of Proposition 37.8.2,
$\square$
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