quantum mechanics - Is there a physical significance to non-normal states of the algebra of observables?

Quantum theory may be formalized in several different ways. Generally, the physical discussion of different states of a quantum system distinguishes pure and mixed states, and then subsumes both in a density operator formalism where a quantum state is given by a density operator $\rho : \mathcal{H}\to\mathcal{H}$ on some Hilbert space $\mathcal{H}$ where $\rho$ is self-adjoint, positive semi-definite and is trace-class with unit trace.

However, in an alternative approach, one starts with the abstract $C^*$-algebra $\mathcal{A}$ of observables and calls any positive linear functional on $\mathcal{A}$ a state. A normal state is then one that can be represented by a density operator on some Hilbert space on which the algebra is represented (there are several equivalent definitions, but this one is the one which makes it obvious that the normal states are the usual pure+mixed states of quantum mechanics).

But, in general, there exist non-normal states at least on some admissible algebras of observables. One can find several vague statements about what those may or may not mean, but I have not yet found a unique satisfactory answer to the question:

Do the non-normal states of the algebra of observables have physical significance, that is, is there an actual quantum system where only considering the normal states leads you to different physical predictions than taking all states?