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''on an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739301.png" /> with an involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739302.png" />''
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A [[Linear functional|linear functional]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739303.png" /> on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739304.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739305.png" /> that satisfies the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739306.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739307.png" />. Positive functionals are important and have been introduced in particular because they are used in the GNS-construction, which is one of the basic methods for examining the structures of Banach <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739308.png" />-algebras. This and its generalizations, for example to weights in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p0739309.png" />-algebras, provide the basis for proving the theorem on the abstract characterization of uniformly-closed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393010.png" />-algebras of operators on a Hilbert space and the theorem on the completeness of a system of irreducible unitary representations of a locally compact group.
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The GNS-construction is a method for constructing a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393011.png" />-representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393012.png" /> of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393013.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393014.png" /> with unit on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393015.png" /> for any positive functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393017.png" />, which is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393018.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393020.png" /> is a certain cyclic vector. The construction is the following: The semi-inner product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393021.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393022.png" />; the corresponding neutral subspace is a left ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393023.png" />, and therefore in the pre-Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393024.png" /> left-multiplication operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393025.png" /> by the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393026.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393027.png" />) are well-defined; the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393028.png" /> are continuous and can be extended to continuous operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393029.png" /> on the completion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393031.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393032.png" /> that takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393033.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393034.png" /> is the required representation, where for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393035.png" /> one can take the image of the unit under the composition of the canonical mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073930/p07393036.png" />.
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''on an algebra  $  A $
 +
with an involution  $  {}  ^ {*} $''
 +
 
 +
A [[Linear functional|linear functional]]  $  f $
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on the  $  {}  ^ {*} $-
 +
algebra  $  A $
 +
that satisfies the condition  $  f( x  ^ {*} x) \geq  0 $
 +
for all  $  x \in A $.
 +
Positive functionals are important and have been introduced in particular because they are used in the GNS-construction, which is one of the basic methods for examining the structures of Banach  $  {}  ^ {*} $-
 +
algebras. This and its generalizations, for example to weights in  $  C  ^ {*} $-
 +
algebras, provide the basis for proving the theorem on the abstract characterization of uniformly-closed  $  {}  ^ {*} $-
 +
algebras of operators on a Hilbert space and the theorem on the completeness of a system of irreducible unitary representations of a locally compact group.
 +
 
 +
The GNS-construction is a method for constructing a $  {}  ^ {*} $-
 +
representation $  \pi _ {f} $
 +
of a $  {}  ^ {*} $-
 +
algebra $  A $
 +
with unit on a Hilbert space $  H _ {f} $
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for any positive functional $  f $
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on $  A $,  
 +
which is such that $  f( x) = \langle  \pi _ {f} ( x) \xi , \xi \rangle $
 +
for all $  x \in A $,  
 +
where $  \xi \in H _ {f} $
 +
is a certain cyclic vector. The construction is the following: The semi-inner product $  \langle  x, y \rangle = f( y  ^ {*} x) $
 +
is defined on $  A $;  
 +
the corresponding neutral subspace is a left ideal $  N _ {f} = \{ {x \in A } : {f( x  ^ {*} x) = 0 } \} $,  
 +
and therefore in the pre-Hilbert space $  A / N _ {f} $
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left-multiplication operators $  L _ {a} $
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by the elements $  a \in A $(
 +
$  L _ {a} ( x + N _ {f} ) = ax + N _ {f} $)  
 +
are well-defined; the operators $  L _ {a} $
 +
are continuous and can be extended to continuous operators $  \overline{L}\; _ {a} $
 +
on the completion $  H _ {f} $
 +
of $  A / N _ {f} $.  
 +
The mapping $  \pi _ {f} $
 +
that takes $  a \in A $
 +
to $  \overline{L}\; _ {a} $
 +
is the required representation, where for $  \xi $
 +
one can take the image of the unit under the composition of the canonical mappings $  A \rightarrow A / N _ {f} \rightarrow H _ {f} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  M.A. Naimark,  "Normed involution rings and their representations"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''12'''  (1948)  pp. 445–480  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Segal,  "Irreducible representations of operator algebras"  ''Bull. Amer. Math. Soc.'' , '''53'''  (1947)  pp. 73–88</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.M. Gel'fand,  M.A. Naimark,  "Normed involution rings and their representations"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''12'''  (1948)  pp. 445–480  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I. Segal,  "Irreducible representations of operator algebras"  ''Bull. Amer. Math. Soc.'' , '''53'''  (1947)  pp. 73–88</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


on an algebra $ A $ with an involution $ {} ^ {*} $

A linear functional $ f $ on the $ {} ^ {*} $- algebra $ A $ that satisfies the condition $ f( x ^ {*} x) \geq 0 $ for all $ x \in A $. Positive functionals are important and have been introduced in particular because they are used in the GNS-construction, which is one of the basic methods for examining the structures of Banach $ {} ^ {*} $- algebras. This and its generalizations, for example to weights in $ C ^ {*} $- algebras, provide the basis for proving the theorem on the abstract characterization of uniformly-closed $ {} ^ {*} $- algebras of operators on a Hilbert space and the theorem on the completeness of a system of irreducible unitary representations of a locally compact group.

The GNS-construction is a method for constructing a $ {} ^ {*} $- representation $ \pi _ {f} $ of a $ {} ^ {*} $- algebra $ A $ with unit on a Hilbert space $ H _ {f} $ for any positive functional $ f $ on $ A $, which is such that $ f( x) = \langle \pi _ {f} ( x) \xi , \xi \rangle $ for all $ x \in A $, where $ \xi \in H _ {f} $ is a certain cyclic vector. The construction is the following: The semi-inner product $ \langle x, y \rangle = f( y ^ {*} x) $ is defined on $ A $; the corresponding neutral subspace is a left ideal $ N _ {f} = \{ {x \in A } : {f( x ^ {*} x) = 0 } \} $, and therefore in the pre-Hilbert space $ A / N _ {f} $ left-multiplication operators $ L _ {a} $ by the elements $ a \in A $( $ L _ {a} ( x + N _ {f} ) = ax + N _ {f} $) are well-defined; the operators $ L _ {a} $ are continuous and can be extended to continuous operators $ \overline{L}\; _ {a} $ on the completion $ H _ {f} $ of $ A / N _ {f} $. The mapping $ \pi _ {f} $ that takes $ a \in A $ to $ \overline{L}\; _ {a} $ is the required representation, where for $ \xi $ one can take the image of the unit under the composition of the canonical mappings $ A \rightarrow A / N _ {f} \rightarrow H _ {f} $.

References

[1] I.M. Gel'fand, M.A. Naimark, "Normed involution rings and their representations" Izv. Akad. Nauk SSSR Ser. Mat. , 12 (1948) pp. 445–480 (In Russian)
[2] I. Segal, "Irreducible representations of operator algebras" Bull. Amer. Math. Soc. , 53 (1947) pp. 73–88
[3] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
How to Cite This Entry:
Positive functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_functional&oldid=48254
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article