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.