fa.functional analysis - Is there a general notion of entropy for the states of a C*algebra?

There are several different definitions of entropy associated with operator algebras. It would be good if you knew which one you were referring to. Do you have a reference? I expect you're talking about the generalization of Kullback-Leibler relative entropy to matrix algebras.

There's a perfectly good definition of entropy for density matrices (positive self-adjoint trace 1 matrices); this is called von Neumann entropy, and although I haven't thought about it I think you should be able to extend it to a definition of entropy for self-adjoint trace 1 operators for II$_1$ factors in von Neumann algebras in the same way that you can extend Shannon entropy to continuous distributions by using differential entropy. In fact, I'd be surprised if that hasn't already been done.