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Character of a C*-algebra

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A non-zero lower semi-continuous semi-finite trace on a -algebra satisfying the following condition (cf. Trace on a -algebra): If is a lower semi-continuous semi-finite trace on and if for all , then for a certain non-negative number and all elements in the closure of the ideal generated by the set . There exists a canonical one-to-one correspondence between the set of quasi-equivalence classes of non-zero factor representations of admitting a trace and the set of characters of defined up to a positive multiplier (cf. Factor representation); this correspondence is established by the formula , , where is the factor representation of admitting the trace . If the trace on is finite, then the character is said to be finite; a finite character is continuous. There exists a canonical one-to-one correspondence between the set of quasi-equivalence classes of non-zero factor representations of finite type of a -algebra and the set of finite characters of with norm 1. If is commutative, then any character of the commutative algebra is a character of the -algebra . If is the group -algebra of a compact group , then the characters of the -algebra are finite, and to such a character with norm 1 there corresponds a normalized character of .

References

[1] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
How to Cite This Entry:
Character of a C*-algebra. A.I. Shtern (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Character_of_a_C*-algebra&oldid=18796
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098