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 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.
↑ 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.

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