The norm of the difference of two states

Question

Suppose $\rho_1$ and $\rho_2$ are two states on a $C^*$-algebra $A$. According to the triangle inequality, we have $\|\rho_1-\rho_2\|\leq 2$.

My question: does there exist a concrete example such that $\|\rho_1-\rho_2\|=2$?

Answer 1

Yes, unless $A$ is one-dimensiomal. If $A$ has projections, it is enough to find two states supported on a projection $p$ and its orthogonal $1-p$ respectively. Then the element with norm one that achieves the maximum is $2p-1=p-(1-p)$.

For instance take $A=M_2(\mathbb C)$, and $$ \rho_1(X)=X_{11},\qquad \rho_2(X)=X_{22}. $$ Then $$ (\rho_1-\rho_2)\bigg(\begin{bmatrix}1&0\\0&-1\end{bmatrix}\bigg)=2. $$ A variation of the above idea can be achieved in $C_0(X)$ for any locally compact Hausdorff space $X$, as long as $X$ is not a singleton. Indeed, taking distinct $r,s\in X$ define $\rho_1(f)=f(r)$, $\rho_2(f)=f(s)$. Then any $f\in C_0(X)$ with $|f|≤1$, $f(r)=1$, $f(s)=-1$, satisfies $(\rho_1-\rho_2)(f)=2$. This last idea can be implemented in any C$^*$-algebra $A$ as long as $A$ is non-trivial. Just take any $a\in A$ with spectrum with at least two points, and do the above construction on $C^*(a)\simeq C(\sigma(a))$.

Note that Hahn-Banach allows you to extend a state from a subalgebra to the full algebra (as a state!), so solving the problem in a subalgebra is enough.