operator algebra In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings. The results obtained in the study o ...

, the Koecher–Vinberg theorem is a reconstruction theorem for real

Jordan algebra In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan a ...

s. It was proved independently by Max Koecher in 1957 and

Ernest Vinberg Ernest Borisovich Vinberg (; 26 July 1937 – 12 May 2020) was a Soviet and Russian mathematician, who worked on Lie groups and algebraic groups, discrete subgroups of Lie groups, invariant theory, and representation theory. He introduced Vinberg ...

in 1961. It provides a

one-to-one correspondence In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ...

between formally real Jordan algebras and so-called domains of positivity. Thus it links

ic and

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

order theoretic views on state spaces of physical systems.

Statement

convex cone In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, C is a cone if x\in C implies sx\in C for e ...

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 In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...

A

with an

inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...

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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...

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