  In this paper, we generalize the notion of the $C$-numerical range of a
matrix to operators in arbitrary tracial von Neumann algebras. For each
self-adjoint operator $C$, the $C$-numerical range of such an operator is
defined; it is a compact, convex subset of $\mathbb{C}$. We explicitly describe
the $C$-numerical ranges of several operators and classes of operators.
