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

## Statement

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.