Koecher–Vinberg theorem

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 C is called regular if a=0 whenever both a and -a are in the closure \overline{C}.

A convex cone C in a vector space A with an inner product has a dual cone \( C^* = \{ a \in A \colon \forall b \in C \langle a,b\rangle > 0 \} \). The cone is called self-dual when \(C=C^* \). It is called homogeneous when to any two points \( a,b \in C \) there is a real linear transformation \( T \colon A \to A \) that restricts to a bijection \( C \to C \) and satisfies 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 is the interior of the 'positive' cone \( A_+ = \{ a^2 \colon a \in A \} \).

Proof

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

References

Koecher, Max (1957). "Positivitatsbereiche im Rn". American Journal of Mathematics 97 (3): 575–596. doi:10.2307/2372563.
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.
J. Faraut and A. Koranyi (1994). Analysis on Symmetric Cones. Oxford University Press.

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