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.
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Statement
A convex cone
A convex cone
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:
Convex cones satisfying these four properties are called domains of positivity or symmetric cones. The domain of positivity associated with a real Jordan algebra
Proof
For a proof, see Koecher (1999) or Faraut & Koranyi (1994).