How to show the norm of the difference of two pure state is 2?

Given two distinct pure state $$\phi_1$$ and $$\phi_2$$ in a commutative unital C$$^*$$-algebra, how do we show $$\|\phi_1-\phi_2\|=2$$?

Of course, we know the norm must be $$\leq 2$$ by the triangle inequality, so all we need to do is to find an element $$a$$ where $$\|\phi_1(a)-\phi_2(a)\|=2$$. I tried doing this but unfortunately, we don't have much information on the norm of $$\|\phi_1(a)-\phi_2(a)\|$$ given some arbitrary $$a$$. Most of the theorems I can currently use are for the converse: given some normal element $$b$$, we know there exists a state $$\omega$$ where $$\omega(b)=\|b\|$$. I could not find much of a way to use this though. Now I have previously proven that states are in fact multiplicative in a commutative unital C$$^*$$-algebra, but I can't think of a good way to apply that. Another approach is just proof by contradiction and assumes as element $$a$$ where $$||\phi_1(a)-\phi_2(a)||=2$$ does not exist and show that either $$\phi_1$$ or $$\phi_2$$ is not a pure state which would imply their corresponding GNS representation is not irreducible so I tried to find an invariant subspace in their GNS representation. Now that also seemed too hard to do so I am still stuck.

• What do the states of a commutative unital $C^\ast$-algebra look like in light of Gel'fand duality? Jun 5, 2022 at 18:26
• I think we just call that the Gelfand Naimark theorem ….so we have a isomorphism with the space of continuous functions on the characters of our initial Calgebra
– Bill
Jun 5, 2022 at 19:11

From the Gelfand-Naimark Theorem it follows that $$A\cong C_0(X)$$ for some locally compact Hausdorf space $$X$$. The state space of $$C_0(X)$$ can be identified with the convex set of (suitably regular) probability measures on $$X$$ and it is not hard to see that the pure states are precisely the point measures. To see that the norm difference of two distinct pure states $$\delta_x$$ and $$\delta_y$$ is two you only need a continuous function $$f \in C_0(X)$$ such that $$f(x) =1$$ and $$f(y)=-1$$ with $$\|f\|_\infty =1$$.