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^[1] and Ernest Vinberg in 1961.^[2] 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 ${\displaystyle C}$ is called regular if ${\displaystyle a=0}$ whenever both ${\displaystyle a}$ and ${\displaystyle -a}$ are in the closure ${\displaystyle {\overline {C}}}$.

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

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 ${\displaystyle A}$ is the interior of the 'positive' cone ${\displaystyle A_{+}=\{a^{2}\colon a\in A\}}$.

Proof

For a proof, see Koecher (1999)^[3] or Faraut & Koranyi (1994).^[4]

References