View source for Koecher–Vinberg theorem

{{Use American English|date = March 2019}}
{{Short description|Theorem of operator algebra}}
In [[operator algebra]], the '''Koecher–Vinberg theorem''' is a reconstruction theorem for real [[Jordan algebra]]s. It was proved independently by [[Max Koecher]] in 1957<ref name="koecher">{{cite journal|last=Koecher|first=Max|title=Positivitatsbereiche im R<sup>n</sup>|journal=American Journal of Mathematics|year=1957|volume=97|issue=3|pages=575–596|doi=10.2307/2372563|jstor=2372563}}</ref> and [[Ernest Vinberg]] in 1961.<ref>{{cite journal|last=Vinberg|first=E. B.|title=Homogeneous Cones|journal=Soviet Math. Dokl.|year=1961|volume=1|pages=787–790}}</ref> It provides a [[one-to-one correspondence]] between [[jordan algebra#Formally real Jordan algebras|formally real Jordan algebras]] and so-called domains of positivity. Thus it links [[operator algebra]]ic and [[Convex function|convex]] [[Order theory|order theoretic]] views on state spaces of physical systems.

==Statement==
A [[convex cone]] <math>C</math> is called ''regular'' if <math>a=0</math> whenever both <math>a</math> and <math>-a</math> are in the closure <math>\overline{C}</math>.

A convex cone <math>C</math> in a [[vector space]] <math>A</math> with an [[inner product]] has a ''dual cone'' <math>C^* = \{ a \in A : \forall b \in C \langle a,b\rangle > 0 \}</math>. The cone is called ''self-dual'' when <math>C=C^*</math>. It is called ''homogeneous'' when to any two points <math>a,b \in C</math> there is a real [[linear transformation]] <math>T \colon A \to A</math> that restricts to a bijection <math>C \to C</math> and satisfies <math>T(a)=b</math>.

The Koecher–Vinberg theorem now states that these properties precisely characterize the positive cones of Jordan algebras.

'''Theorem''': There is a one-to-one correspondence between [[jordan algebra#Formally real Jordan algebras|formally real Jordan algebras]] and convex cones that are:
* open;
* regular;
* homogeneous;
* self-dual.

Convex cones satisfying these four properties are called ''domains of positivity'' or ''[[Symmetric cone|symmetric cones]]''. The domain of positivity associated with a real Jordan algebra <math>A</math> is the interior of the 'positive' cone <math>A_+ = \{ a^2 \colon a \in A \}</math>.

==Proof==
For a proof, see {{harvtxt|Koecher|1999}}<ref name=minnesota>{{cite book|last=Koecher|first=Max|title=The Minnesota Notes on Jordan Algebras and Their Applications|year=1999|publisher=Springer|isbn=3-540-66360-6|url=https://books.google.com/books?id=RHrdf06-vZ0C}}
</ref> or {{harvtxt|Faraut|Koranyi|1994}}.<ref name=farautkoranyi>{{cite book|first1=J.|last1=Faraut|author2-link=Ádám Korányi|first2=A.|last2=Koranyi|title=Analysis on Symmetric Cones|year=1994|publisher=Oxford University Press}}</ref>

==References==
{{reflist}}

{{DEFAULTSORT:Koecher-Vinberg theorem}}
[[Category:Non-associative algebras]]
[[Category:Theorems in algebra]]