In operator algebra, the Koecher窶天inberg 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.
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窶天inberg 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 is the interior of the 'positive' cone .
For a proof, see Koecher (1999) [3] or Faraut & Koranyi (1994). [4]
In operator algebra, the Koecher窶天inberg 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.
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窶天inberg 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 is the interior of the 'positive' cone .
For a proof, see Koecher (1999) [3] or Faraut & Koranyi (1994). [4]