# What exactly is meant by a semicontinuous, semifinite trace on a C* algebra?

I see phrases like "$\tau$ is a semicontinuous, semifinite trace on a C* algebra $A$" thrown around a lot in papers without any extra qualification. So, I gather this terminology is standard? I've never been exactly sure what it means though. I am pretty sure that, in the case $A = C_0(X)$ for $X$ a locally compact Hausdorff space, this is just supposed to mean "$\tau$ is integration against a positive, regular, Borel measure on $X$". My strategy has always been to read any and all arguments involving such $\tau$ while secretly assuming $A$ is commutative. Could somebody please provide a precise definition of a "semicontinuous, semifinite trace on a C* algebra" or, better yet, point me toward a nice reference for such things? Thanks.

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The EOM entry seems to answer that question and refers to Dixmier. Nevertheless it would be nice to have something slightly ... less condensed as an answer. – t.b. Aug 4 '12 at 21:46

One thing which seems to be done is define a weight on a (non-unital) C*-algebra $A$ to be a mapping $\phi$ from the positive cone $A_+$ of $A$ to $[0,\infty]$ such that $\phi(x+y) = \phi(x) + \phi(y)$ and $\phi(\lambda x) = \lambda \phi(x)$ are satisfied for $x,y \in A_+$, $\lambda \in [0,\infty)$ (with the usual measure-theoretic convention that $0 \cdot \infty = 0$). One then says that $\phi$ is lower semi-continuous if $\phi(x) \leq \liminf \phi(x_i)$ whenever $x_i \to x$ in $A_+$ (a sort of Fatou's lemma-type condition). I think one says $\phi$ is semi-finite to mean that $\{x \in A_+ : \phi(x) < \infty\}$ is dense in $A_+$, but the terminology densely-defined would certainly be preferable for this. A densely-defined, lower semicontinuous trace would then be a densely-defined, lower semi-continuous weight $\tau$ on $A$ which is invariant under the inner automorphism group of $A$ i.e. $\phi(uxu^*) = \phi(x)$ for all $x \in A_+$ and all unitaries $u$ in the minimal unitization of $A$.