# 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)
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=53350
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article