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.
Statement A convex cone C {\displaystyle C} is called regular if a = 0 {\displaystyle a=0} whenever both a {\displaystyle a} and − a {\displaystyle -a} are in the closure C ¯ {\displaystyle {\overline {C}}} . A convex cone C {\displaystyle C} in a vector space A {\displaystyle A} with an inner product has a dual cone C ∗ = { a ∈ A : ∀ b ∈ C ⟨ a , b ⟩ > 0 } {\displaystyle C^{*}=\{a\in A:\forall b\in C\langle a,b\rangle >0\}} . The cone is called self-dual when C = C ∗ {\displaystyle C=C^{*}} . It is called homogeneous when to any two points a , b ∈ C {\displaystyle a,b\in C} there is a real linear transformation T : A → A {\displaystyle T\colon A\to A} that restricts to a bijection C → C {\displaystyle C\to C} and satisfies T ( a ) = b {\displaystyle T(a)=b} . 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 A {\displaystyle A} is the interior of the 'positive' cone A + = { a 2 : a ∈ A } {\displaystyle A_{+}=\{a^{2}\colon a\in A\}} .
Proof For a proof, see Koecher (1999) or Faraut & Koranyi (1994).
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