# 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 convexTemplate:Dn order theoretic views on state spaces of physical systems.

## Statement

A convex cone is called *regular* if whenever both and are in the closure .

A convex cone in a vector space with an inner product has a *dual cone* . The cone is called *self-dual* when . It is called *homogeneous* when to any two points there is a real linear transformation that restricts to a bijection and satisfies .

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 is the interior of the 'positive' cone .

## Proof

For a proof, see^{[3]} or.^{[4]}