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 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 $C$ 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 Koecher (1999) or Faraut & Koranyi (1994).