= Maximal ideals of a commutative complex unital Banach algebra are character kernels
{title2=$\Phi_A\longleftrightarrow\mathcal M_A,\quad\phi\mapsto\ker\phi$}
In a commutative complex unital <Banach algebra>, a <maximal ideal> is closed: its closure remains proper, since an element within distance less than one of the identity is invertible by the <Neumann series>. The quotient is a complex division <Banach algebra>, hence is the complex <field> by the <Gelfand-Mazur theorem>. Its quotient map is a <character of an algebra>. Conversely a nonzero multiplicative complex <linear functional> has value one at the identity, is onto the complex <field>, and has maximal kernel. It is automatically continuous because its value at each element belongs to that element's spectrum. The kernel determines the <algebra character> uniquely since the quotient is one-dimensional.
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