Koecher–Vinberg theorem

In operator algebra, the Koecher–Vinberg theorem is a reconstruction theorem for real Jordan algebras. It was proved independently by Max Koecher in 1957{{cite journal|last=Koecher|first=Max|title=Positivitatsbereiche im Rⁿ|journal=American Journal of Mathematics|year=1957|volume=97|issue=3|pages=575–596|doi=10.2307/2372563|jstor=2372563}} and Ernest Vinberg in 1961.{{cite journal|last=Vinberg|first=E. B.|title=Homogeneous Cones|journal=Soviet Math. Dokl.|year=1961|volume=1|pages=787–790}} It provides a one-to-one correspondence between formally real Jordan algebras and so-called domains of positivity. Thus it links operator algebraic and convex order theoretic views on state spaces of physical systems.

Statement

A convex cone $C$ is called regular if $a=0$ whenever both $a$ and $-a$ are in the closure $\overline{C}$ .

A convex cone $C$ in a vector space $A$ with an inner product has a dual cone $C^* = \{ a \in A : \forall b \in C \langle a,b\rangle > 0 \}$ . The cone is called self-dual when $C=C^*$ . It is called homogeneous when to any two points $a,b \in C$ there is a real linear transformation $T \colon A \to A$ that restricts to a bijection $C \to C$ and satisfies $T(a)=b$ .

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 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 cones. The domain of positivity associated with a real Jordan algebra $A$ is the interior of the 'positive' cone $A_+ = \{ a^2 \colon a \in A \}$ .

Proof

For a proof, see {{harvtxt|Koecher|1999}}{{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}}

or {{harvtxt|Faraut|Koranyi|1994}}.{{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}}

References

Category:Non-associative algebras

Category:Theorems in algebra