Theorem of operator algebra
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 formula_1 is called "regular" if formula_2 whenever both formula_3 and formula_4 are in the closure formula_5.
A convex cone formula_1 in a vector space formula_7 with an inner product has a "dual cone" formula_8. The cone is called "self-dual" when formula_9. It is called "homogeneous" when to any two points formula_10 there is a real linear transformation formula_11 that restricts to a bijection formula_12 and satisfies formula_13.
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 formula_7 is the interior of the 'positive' cone formula_15.
Proof.
For a proof, see or .
References.
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