Koecher–Vinberg theorem

Short description: Theorem of operator algebra

In operator algebra, the Koecher–Vinberg theorem is a reconstruction theorem for real Jordan algebras. It was proved independently by Max Koecher in 1957 and Ernest Vinberg in 1961. 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 [math]\displaystyle{ C }[/math] is called regular if [math]\displaystyle{ a=0 }[/math] whenever both [math]\displaystyle{ a }[/math] and [math]\displaystyle{ -a }[/math] are in the closure [math]\displaystyle{ \overline{C} }[/math].

A convex cone [math]\displaystyle{ C }[/math] in a vector space [math]\displaystyle{ A }[/math] with an inner product has a dual cone [math]\displaystyle{ C^* = \{ a \in A : \forall b \in C \langle a,b\rangle \gt 0 \} }[/math]. The cone is called self-dual when [math]\displaystyle{ C=C^* }[/math]. It is called homogeneous when to any two points [math]\displaystyle{ a,b \in C }[/math] there is a real linear transformation [math]\displaystyle{ T \colon A \to A }[/math] that restricts to a bijection [math]\displaystyle{ C \to C }[/math] and satisfies [math]\displaystyle{ 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 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 [math]\displaystyle{ A }[/math] is the interior of the 'positive' cone [math]\displaystyle{ A_+ = \{ a^2 \colon a \in A \} }[/math].

Proof

For a proof, see (Koecher 1999) or (Faraut Koranyi).