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