Gelfand–Mazur theorem

Every complex unital Banach algebra in which every nonzero element is invertible is isometrically isomorphic, as a unital complex algebra, to the complex numbers.

Connect

Spectrum of an element: The set of complex numbers λ for which λ1 − a is not invertible in a unital algebra. The theorem’s proof applies nonemptiness of this set to each element.
Unital algebra: An algebra with a multiplicative identity element. The identity lets the spectrum be defined through λ1 − a.
Real division algebra: A division algebra whose scalar field is the real numbers. Finite-dimensional real division algebras include ℝ, ℂ, and the quaternions.
Gelfand representation: A representation of a commutative Banach algebra as an algebra of functions on its space of characters. Its characters yield scalar-valued quotients that connect Banach algebras to ℂ.
Banach algebra: A complete normed algebra whose norm is submultiplicative. Completeness supplies the spectral theorem used in the proof.
Division algebra: An algebra in which every nonzero element has a multiplicative inverse. This condition makes every spectral value of an element equal to that element’s scalar value.
Quaternion algebra: A four-dimensional real associative division algebra with noncommutative multiplication. It is a Banach division algebra over ℝ, but not a complex one.
Maximal ideal: A proper ideal not strictly contained in any other proper ideal. Quotients by maximal ideals produce complex Banach division algebras.
Liouville's theorem: A bounded entire function on the complex plane is constant. Resolvent estimates and complex analysis underpin spectral nonemptiness.
Complex Banach space: A complete normed vector space over the complex numbers. The algebra’s complex scalar structure enables complex-analytic proof methods.