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
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 Koecher (1999)[3] or Faraut & Koranyi (1994).[4]
References
- ↑ 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.