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Character of a representation of an associative algebra

From Encyclopedia of Mathematics
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A function on the associative algebra defined by the formula for , where is a representation of and is a linear functional defined on some ideal in , satisfying the condition for all , . If the representation is finite-dimensional or if the algebra contains a non-zero finite-dimensional operator, then for one usually considers the trace of the operator. Let be a -algebra, a representation of the -algebra such that the von Neumann algebra generated by is a factor of semi-finite type; let be a faithful normal semi-finite trace on and let be a linear extension of to an ideal . If the set is non-zero, then the formula , , determines a character of the representation of the algebra whose restriction to is a character of the -algebra (cf. Character of a -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 -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=46314
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article