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 or.