# 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) |

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Character of a C*-algebra.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Character_of_a_C*-algebra&oldid=55347