Riesz decomposition property

Let $( E , C )$ be a partially ordered vector space, [a5], i.e. $E$ is a real vector space with a convex cone $C$ defining the partial order by $x \succ y$ if and only if $x - y \in C$. For $x \prec y$, the corresponding interval is $[ x , y ] = \{ u \in E : x \prec u \prec y \}$.

The (partially) ordered vector space $( E , C )$ has the Riesz decomposition property if $[ 0 , u ] + [ 0 , v ] = [ 0 , u + v ]$ for all $u , v \in C$, or, equivalently, if $[ x _ { 1 } , y _ { 1 } ] + [ x _ { 2 } , y _ { 2 } ] = [ x _ { 1 } + x _ { 2 } , y _ { 1 } + y _ { 2 } ]$ for all $x _ { 1 } \prec y _ { 1 }$, $x _ { 2 } \prec y _ { 2 }$.

A Riesz space (or vector lattice) automatically has the Riesz decomposition property.

Terminology on this concept varies a bit: in [a2] the property is referred to as the dominated decomposition property, while in [a3] it is called the decomposition property of F. Riesz.

The Riesz decomposition property and the Riesz decomposition theorem are quite different (although there is a link) and, in fact, the property does turn up as an axiom used in axiomatic potential theory (see also Potential theory, abstract), see [a1], where it is called the axiom of natural decomposition.

There is a natural non-commutative generalization to the setting of $C ^ { * }$-algebras, as follows, [a4]. Let $x$, $y$, $z$ be elements of a $C ^ { * }$-algebra $A$. If $x ^ { * } x \leq y y ^ { * } + z z ^ { * }$, then there are $u , v \in A$ such that $u ^ { * } u \leq y ^ { * } y$, $v ^ { * } v \leq x ^ { * } x$ and $x x ^ { * } = u u ^ { * } + v v ^ { * }$.

References

[a1]	C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972) pp. 104
[a2]	W.A.J. Luxemburg, A.C. Zaanen, "Riesz spaces" , I , North-Holland (1971) pp. 73
[a3]	P. Meyer-Nieberg, "Banach lattices" , Springer (1971) pp. 3, Thm. 1.1.1
[a4]	G.K. Pedersen, "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press (1979) pp. 14
[a5]	Y.-Ch. Wong, K.-F. Ng, "Partially ordered topological vector spaces" , Oxford Univ. Press (1973) pp. 9