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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215801.png" /> on the associative algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215802.png" /> defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215803.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215804.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215805.png" /> is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215807.png" /> is a linear functional defined on some ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215808.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c0215809.png" />, satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158012.png" />. If the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158013.png" /> is finite-dimensional or if the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158014.png" /> contains a non-zero finite-dimensional operator, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158015.png" /> one usually considers the trace of the operator. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158016.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158017.png" />-algebra, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158018.png" /> a representation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158019.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158020.png" /> such that the [[Von Neumann algebra|von Neumann algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158021.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158022.png" /> is a [[Factor|factor]] of semi-finite type; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158023.png" /> be a faithful normal semi-finite trace on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158024.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158025.png" /> be a linear extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158026.png" /> to an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158027.png" />. If the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158028.png" /> is non-zero, then the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158030.png" />, determines a character of the representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158031.png" /> whose restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158032.png" /> is a character of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158033.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158034.png" /> (cf. [[Character of a C*-algebra|Character of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158035.png" />-algebra]]). In many cases the character of a representation of an algebra determines the representation uniquely, up to a certain equivalence relation; for example, the character of an irreducible finite-dimensional representation determines the representation uniquely up to equivalence; the character of a factor representation of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158036.png" />-algebra admitting a trace determines the representation uniquely up to quasi-equivalence.
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A function  $  \phi $
 +
on the associative algebra $  A $
 +
defined by the formula $  \phi ( x) = \chi ( \pi ( x)) $
 +
for $  x \in A $,  
 +
where $  \pi $
 +
is a representation of $  A $
 +
and $  \chi $
 +
is a linear functional defined on some ideal $  I $
 +
in $  \pi ( A) $,  
 +
satisfying the condition $  \chi ( ab) = \chi ( ba) $
 +
for all $  a \in I $,  
 +
$  b \in \pi ( A) $.  
 +
If the representation $  \pi $
 +
is finite-dimensional or if the algebra $  \pi ( A) $
 +
contains a non-zero finite-dimensional operator, then for $  \chi $
 +
one usually considers the trace of the operator. Let $  A $
 +
be a $  C  ^ {*} $-
 +
algebra, $  \pi $
 +
a representation of the $  C  ^ {*} $-
 +
algebra $  A $
 +
such that the [[Von Neumann algebra|von Neumann algebra]] $  \mathfrak A $
 +
generated by $  \pi ( A) $
 +
is a [[Factor|factor]] of semi-finite type; let $  \chi  ^  \prime  $
 +
be a faithful normal semi-finite trace on $  \mathfrak A $
 +
and let $  \chi $
 +
be a linear extension of $  \chi  ^  \prime  $
 +
to an ideal $  \mathfrak M _ {\chi  ^  \prime  } $.  
 +
If the set $  \{ {x } : {x \in A,  \chi  ^  \prime  ( \pi ( x)) < + \infty } \} $
 +
is non-zero, then the formula $  \phi ( x) = \chi ( \pi ( x)) $,  
 +
$  x \in A $,  
 +
determines a character of the representation of the algebra $  A $
 +
whose restriction to $  A  ^ {+} $
 +
is a character of the $  C  ^ {*} $-
 +
algebra $  A $(
 +
cf. [[Character of a C*-algebra|Character of a $  C  ^ {*} $-
 +
algebra]]). In many cases the character of a representation of an algebra determines the representation uniquely, up to a certain equivalence relation; for example, the character of an irreducible finite-dimensional representation determines the representation uniquely up to equivalence; the character of a factor representation of a $  C  ^ {*} $-
 +
algebra admitting a trace determines the representation uniquely up to quasi-equivalence.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158037.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.A. Kirillov,  "Elements of the theory of representations" , Springer  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.W. Curtis,  I. Reiner,  "Representation theory of finite groups and associative algebras" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021580/c02158037.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>

Revision as of 16:43, 4 June 2020


A function $ \phi $ on the associative algebra $ A $ defined by the formula $ \phi ( x) = \chi ( \pi ( x)) $ for $ x \in A $, where $ \pi $ is a representation of $ A $ and $ \chi $ is a linear functional defined on some ideal $ I $ in $ \pi ( A) $, satisfying the condition $ \chi ( ab) = \chi ( ba) $ for all $ a \in I $, $ b \in \pi ( A) $. If the representation $ \pi $ is finite-dimensional or if the algebra $ \pi ( A) $ contains a non-zero finite-dimensional operator, then for $ \chi $ one usually considers the trace of the operator. Let $ A $ be a $ C ^ {*} $- algebra, $ \pi $ a representation of the $ C ^ {*} $- algebra $ A $ such that the von Neumann algebra $ \mathfrak A $ generated by $ \pi ( A) $ is a factor of semi-finite type; let $ \chi ^ \prime $ be a faithful normal semi-finite trace on $ \mathfrak A $ and let $ \chi $ be a linear extension of $ \chi ^ \prime $ to an ideal $ \mathfrak M _ {\chi ^ \prime } $. If the set $ \{ {x } : {x \in A, \chi ^ \prime ( \pi ( x)) < + \infty } \} $ is non-zero, then the formula $ \phi ( x) = \chi ( \pi ( x)) $, $ x \in A $, determines a character of the representation of the algebra $ A $ whose restriction to $ A ^ {+} $ is a character of the $ C ^ {*} $- algebra $ A $( cf. Character of a $ C ^ {*} $- algebra). In many cases the character of a representation of an algebra determines the representation uniquely, up to a certain equivalence relation; for example, the character of an irreducible finite-dimensional representation determines the representation uniquely up to equivalence; the character of a factor representation of a $ C ^ {*} $- algebra admitting a trace determines the representation uniquely up to quasi-equivalence.

References

[1] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[2] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962)
[3] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
How to Cite This Entry:
Character of a representation of an associative algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Character_of_a_representation_of_an_associative_algebra&oldid=19265
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article