Difference between revisions of "Riesz decomposition property"
From Encyclopedia of Mathematics
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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 | + | 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>< | + | <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=50384
Riesz decomposition property. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_decomposition_property&oldid=50384
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article