# 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) |

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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=53350