> [!definition] > > Let $\alg$ be a commutative unital [[Banach Algebra|Banach algebra]]. A **multiplicative [[Linear Functional|functional]]** is a *non-zero* Banach algebra homomorphism from $\alg$ to $\complex$. > [!theorem] > > Let $h$ be a multiplicative functional on $\alg$, then > 1. $h(e) = 1$ > 2. If $x$ is invertible in $\alg$, then $h(x) \ne 0$. > 3. $\abs{h(x)} \le \norm{x}$. > > In particular, $\sigma(\alg) \subset \ol{B(0, 1)} \subset \alg^*$. Thus we place the [[Weak Topology|weak*-topology]] on $\sigma(\alg)$, making it a [[Compactness|compact]] [[Hausdorff Space|Hausdorff space]]. > > Meaning that the set of multiplicative functionals is a [[Closed Set|closed]] subset of the closed unit ball of $\alg^*$. Inheriting the [[Operator Topologies|weak* topology]] from $\alg^*$, the set of multiplicative functionals forms a compact Hausdorff space. > > *Proof*. Since $h(ex) = h(e)h(x) = h(x)$, $h(e) = 1$. If $x$ is invertible, then $h(xx^{-1}) = h(x)h(x^{-1}) = 1$. Lastly, if $\abs{\lambda} > \norm{x}$, then $\lambda e - x$ is invertible, and $h(\lambda e - x) = \lambda - h(x) \ne 0$. With proper orientation, $\abs{h(x)} \le \norm{x}$. > [!theorem] > > Let $\alg$ be a commutative unital Banach algebra, then the map $h \to \ker{(h)}$ is a bijection between the set of multiplicative functionals and the [[Ideal (Algebra over Field)|maximal ideals]] of $\alg$. The set hence will be called the **spectrum** of $\alg$. > > *Proof*. If $h$ is a multiplicative functional, then $\ker(h)$ is a proper, closed ideal because it has codimension one. If $\ker(g) = \ker(h)$, then $g$ and $h$ are linearly dependent. Since they agree on $e$, they are exactly equal. > > On the other hand, let $\cm$ be a maximal ideal, then $\cm$ is closed and we can take the quotient. Let $\pi: \alg \to \alg/\cm$ be the canonical map. Since $\cm$ is maximal, $\alg/\cm$ has no non-trivial ideals, and is isomorphic to $\complex$. Let $\phi$ be the isomorphism from $\alg/\cm$ to $\complex$, then $\phi \circ \pi$ is the desired multiplicative functional.