View source for Riesz decomposition property

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301201.png" /> be a partially ordered vector space, [[#References|[a5]]], i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301202.png" /> is a real [[Vector space|vector space]] with a convex cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301203.png" /> defining the [[Partial order|partial order]] by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301204.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301205.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301206.png" />, the corresponding interval is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301207.png" />.

The (partially) ordered vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301208.png" /> has the Riesz decomposition property if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r1301209.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012010.png" />, or, equivalently, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012011.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012013.png" />.

A [[Riesz space|Riesz space]] (or [[Vector lattice|vector lattice]]) automatically has the Riesz decomposition property.

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

The Riesz decomposition property and the [[Riesz decomposition theorem|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|Potential theory, abstract]]), see [[#References|[a1]]], where it is called the axiom of natural decomposition.

There is a natural non-commutative generalization to the setting of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012014.png" />-algebras, as follows, [[#References|[a4]]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012017.png" /> be elements of a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012018.png" />-algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012019.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012020.png" />, then there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012021.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012024.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)  pp. 104</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W.A.J. Luxemburg,   A.C. Zaanen,   "Riesz spaces" , '''I''' , North-Holland  (1971)  pp. 73</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Meyer-Nieberg,   "Banach lattices" , Springer  (1971)  pp. 3, Thm. 1.1.1</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.K. Pedersen,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130120/r13012025.png" />-algebras and their automorphism groups" , Acad. Press  (1979)  pp. 14</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  Y.-Ch. Wong,   K.-F. Ng,   "Partially ordered topological vector spaces" , Oxford Univ. Press  (1973)  pp. 9</TD></TR></table>