View source for Character of a representation of an associative algebra

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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====
<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,   "$C^\star$ algebras" , North-Holland  (1977)  (Translated from French)</TD></TR>
</table>