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Difference between revisions of "Riesz decomposition property"

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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" />.
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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" />.
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Let $( E , C )$ be a partially ordered vector space, [[#References|[a5]]], i.e. $E$ is a real [[Vector space|vector space]] with a convex cone $C$ defining the [[Partial order|partial order]] by $x \succ y$ if and only if $x - y \in C$. For $x \prec y$, the corresponding interval is $[ x , y ] = \{ u \in E : x \prec u \prec y \}$.
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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|Riesz space]] (or [[Vector lattice|vector lattice]]) automatically has the Riesz decomposition property.
 
A [[Riesz space|Riesz space]] (or [[Vector lattice|vector lattice]]) automatically has the Riesz decomposition property.
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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.
 
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" />.
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There is a natural non-commutative generalization to the setting of $C ^ { * }$-algebras, as follows, [[#References|[a4]]]. Let $x$, $y$, $z$ be elements of a [[C*-algebra|$C ^ { * }$-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====
 
====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>
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<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,  "$C ^ { * }$-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>

Latest revision as of 17:00, 1 July 2020

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
How to Cite This Entry:
Riesz decomposition property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_decomposition_property&oldid=18014
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article