In
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...
, the Gelfand–Mazur theorem is a
theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...
named after
Israel Gelfand
Israel Moiseevich Gelfand, also written Israïl Moyseyovich Gel'fand, or Izrail M. Gelfand (, , ; – 5 October 2009) was a prominent Soviet and American mathematician, one of the greatest mathematicians of the 20th century, biologist, teache ...
and
Stanisław Mazur
Stanisław Mieczysław Mazur (; 1 January 1905 – 5 November 1981) was a Polish mathematician and a member of the Polish Academy of Sciences.
Mazur made important contributions to geometrical methods in linear and nonlinear functional analysis ...
which states that a
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach sp ...
with unit over the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s in which every nonzero element is
invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that ...
is
isometrically isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
s, i. e., the only complex Banach algebra that is a
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
is the complex numbers
.
The theorem follows from the fact that the
spectrum
A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of any element of a complex Banach algebra is nonempty: for every element
of a complex Banach algebra
there is some complex number
such that
is not invertible. This is a consequence of the complex-analyticity of the
resolvent function. By assumption,
. So
. This gives an isomorphism from
to
.
The theorem can be strengthened to the claim that there are (up to isomorphism) exactly three real Banach division algebras: the field of reals
, the field of complex numbers
, and the division algebra of
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternion ...
. This result was proved first by Stanisław Mazur alone, but it was published in France without a proof, when the author refused the editor's request to shorten his proof. Gelfand (independently) published a proof of the complex case a few years later.
References
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{{DEFAULTSORT:Gelfand-Mazur Theorem
Banach algebras
Theorems in functional analysis