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.
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
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: Convex cones satisfying these four properties are called domains of positivity or symmetric cones.
The domain of positivity associated with a real Jordan algebra
For a proof, see Koecher (1999)[3] or Faraut & Koranyi (1994).