Normal and locally normal states

Abstract

It is shown that if$\mathfrak{A}$ is an irreducibleC* algebra on a Hilbert space ℋ andN is the set of normal states of$\mathfrak{A}$ then the weak and uniform topologies onN coincide and are identical to the weak*-$\mathfrak{A}$ topology for each$\mathfrak{A} \supset \mathfrak{L}\mathfrak{C}$(ℋ). It is further shown that all weak* topologies coincide with the uniform topology on the set of normal states with finite energy or with finite conditional entropy. A number of continuity properties of the spectra of density matrices, the mean energy, and the conditional entropy are also derived. The extension of these results to locally normal states is indicated and it is established that locally normal factor states are characterized by a doubly uniform clustering property.