Character of a representation of an associative algebra

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, "$C^\star$ algebras" , North-Holland (1977) (Translated from French)