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 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 $\overline$.

A convex cone $C$ in a vector space $A$ with an inner product has a ''dual cone'' $C^* = \$. 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_+ = \$.

Proof

For a proof, see
or .

References

{{DEFAULTSORT:Koecher-Vinberg theorem
Non-associative algebras
Theorems in algebra