Koecher–Vinberg theorem

From formulasearchengine
Jump to navigation Jump to search

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.


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 .


For a proof, see[3] or.[4]


  1. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  2. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
  3. {{#invoke:citation/CS1|citation |CitationClass=book }}
  4. {{#invoke:citation/CS1|citation |CitationClass=book }}