Character of a C*-algebra

A non-zero lower semi-continuous semi-finite trace $ f $ on a $ C ^ {*} $- algebra $ A $ satisfying the following condition (cf. Trace on a $ C ^ {*} $- algebra): If $ \phi $ is a lower semi-continuous semi-finite trace on $ A $ and if $ \phi ( x) \leq f ( x) $ for all $ x \in A ^ {+} $, then $ \phi ( x) = \lambda f ( x) $ for a certain non-negative number $ \lambda $ and all elements $ x \in A ^ {+} $ in the closure of the ideal $ \mathfrak N _ {f} $ generated by the set $ \{ {x } : {x \in A ^ {+} , f ( x) < + \infty } \} $. There exists a canonical one-to-one correspondence between the set of quasi-equivalence classes of non-zero factor representations of $ A $ admitting a trace and the set of characters of $ A $ defined up to a positive multiplier (cf. Factor representation); this correspondence is established by the formula $ f ( x) = \chi ( \pi ( x)) $, $ x \in A $, where $ \pi $ is the factor representation of $ A $ admitting the trace $ \chi $. If the trace $ f $ on $ A $ 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 $ C ^ {*} $- algebra $ A $ and the set of finite characters of $ A $ with norm 1. If $ A $ is commutative, then any character of the commutative algebra $ A $ is a character of the $ C ^ {*} $- algebra $ A $. If $ A $ is the group $ C ^ {*} $- algebra of a compact group $ G $, then the characters of the $ C ^ {*} $- algebra $ A $ are finite, and to such a character with norm 1 there corresponds a normalized character of $ G $.

References

[1]	J. Dixmier, "$C^*$ algebras" , North-Holland (1977) (Translated from French)