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

Question

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.

Answer 1

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 $A$.

Many references could be given for this material. There is one which happens to be sitting on my desk. See the appendix of Phillips and Raeburn, An Index Theorem for Toeplitz Operators with Noncommutative Symbol Space.