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.
Statement
editA 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
editFor a proof, see Koecher (1999)[3] or Faraut & Koranyi (1994).[4]
References
edit- ^ Koecher, Max (1957). "Positivitatsbereiche im Rn". American Journal of Mathematics. 97 (3): 575–596. doi:10.2307/2372563. JSTOR 2372563.
- ^ Vinberg, E. B. (1961). "Homogeneous Cones". Soviet Math. Dokl. 1: 787–790.
- ^ Koecher, Max (1999). The Minnesota Notes on Jordan Algebras and Their Applications. Springer. ISBN 3-540-66360-6.
- ^ Faraut, J.; Koranyi, A. (1994). Analysis on Symmetric Cones. Oxford University Press.