# Trace on a C*-algebra

$A$

An additive functional $f$ on the set $A ^ {+}$ of positive elements of $A$ that takes values in $[ 0, + \infty ]$, is homogeneous with respect to multiplication by positive numbers and satisfies the condition $f ( xx ^ {*} ) = f ( x ^ {*} x)$ for all $x \in A$. A trace $f$ is said to be finite if $f ( x) < + \infty$ for all $x \in A ^ {+}$, and semi-finite if $f ( x) = \sup \{ {f ( y) } : {y \in A, y \leq x, f( y) < + \infty } \}$ for all $x \in A ^ {+}$. The finite traces on $A$ are the restrictions to $A ^ {+}$ of those positive linear functionals $\phi$ on $A$ such that $\phi ( xy) = \phi ( yx)$ for all $x, y \in A$. Let $f$ be a trace on $A$, let $\mathfrak N _ {f}$ be the set of elements $x \in A$ such that $f ( xx ^ {*} ) < + \infty$, and let $\mathfrak M _ {f}$ be the set of linear combinations of products of pairs of elements of $\mathfrak N _ {f}$. Then $\mathfrak N _ {f}$ and $\mathfrak M _ {f}$ are self-adjoint two-sided ideals of $A$, and there is a unique linear functional $\phi$ on $\mathfrak M _ {f}$ that coincides with $f$ on $\mathfrak M _ {f} \cap A ^ {+}$. Let $f$ be a lower semi-continuous semi-finite trace on a $C ^ {*}$- algebra $A$. Then the formula $s ( x, y) = \phi ( y ^ {*} x)$ defines a Hermitian form on $\mathfrak N _ {f}$, with respect to which the mapping $\lambda _ {f} ( x): x \mapsto xy$ of $\mathfrak N _ {f}$ into itself is continuous for any $x \in A$. Put $N _ {f} = \{ {x \in \mathfrak N _ {f} } : {s ( x, x) = 0 } \}$, and let $H _ {f}$ be the completion of the quotient space $\mathfrak N _ {f} /N _ {f}$ with respect to the scalar product defined by the form $s$. By passing to the quotient space and subsequent completion, the operators $\lambda _ {f} ( x)$ determine certain operators $\pi _ {f} ( x)$ on the Hilbert space $H _ {f}$, and the mapping $x \mapsto \pi _ {f} ( x)$ is a representation of the $C ^ {*}$- algebra $A$ in $H _ {f}$. The mapping $f \mapsto \pi _ {f}$ establishes a one-to-one correspondence between the set of lower semi-continuous semi-finite traces on $A$ and the set of representations of $A$ with a trace, defined up to quasi-equivalence.

#### References

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