Koecher–Vinberg theorem - Alchetron, the free social encyclopedia

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 is called regular if a = 0 whenever both a and − a are in the closure C ¯ .

A convex cone C in a vector space A with an inner product has a dual cone C ∗ = { a ∈ A : ∀ b ∈ C ⟨ a , b ⟩ > 0 } . The cone is called self-dual when C = C ∗ . It is called homogeneous when to any two points a , b ∈ C there is a real linear transformation T : A → A that restricts to a bijection C → 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 : a ∈ A } .

Proof

For a proof, see Koecher (1999) or Faraut & Koranyi (1994).

References

Koecher–Vinberg theorem Wikipedia

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Contents

Statement

Proof

References