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 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 or .
References