Definition of normal linear functional

Question

I am confused about the definition of normal linear functional.

In mamy reference books, when we define normal linear functionals , we require that the linear functional is positive. But in Kadison's book, the normal linear functional may not be positive.(See the theorem in the screenshot).

I don't understand why we need the positivity of linear functional when we defining the normality in many cases.

Answer 1

Kadison-Ringrose are very clear on this, in the section you are quoting. In previous sections, "normal" is only applied to states.

But in section 7.4, they state:

7.4.1. DEFINITION. If $\mathscr R$ is a von Neumann algebra acting on the Hilbert space $\mathscr H$, we denote by $\mathscr R_\#$ the linear space of linear functionals on $\mathscr R$ that are weak-operator continuous on the unit ball of $\mathscr R$ (the "normal" linear functionals on $\mathscr R$). We refer to $\mathscr R_\#$ are predual of $\mathscr R$.

And in the page right before the one you quoted:

Applied to ultraweakly continuous states, Proposition 7.4.5 constitutes an additional condition to Theorem 7.1.12, The terms "normal" and "ultraweakly continuous" are now equally applicable to functionals on $\mathscr R$.