I came across this in a blog post.
Let $A$ be a subset of $n \times n$ complex matrices (which can be viewed as a $C^*$-algebra). Then they write $\|A\|$ (which I assume to be operator norm) as $\max_{s\ \in \text{ states on A}} s(A)$. I think states in a $C^*$-algebra are positive linear functionals with norm at most 1, but I wasn't sure why the operator norm of $A$ can be viewed as a maximum over states on the algebra operating on $A$? Could someone clarify that?
In the usual way of identifying an algebra of matrices (or more generally operators on a Hilbert space) as a $C^*$-algebra, the norm is the operator norm. However, one would usually take the scalars to be complex, so I question whether you really want real matrices here.
Your statement about states is wrong. For the $C^*$ algebra of $n \times n$ complex matrices, the states are convex combinations of the pure states corresponding to unit vectors: $$ \varphi_v(A) = v^* A v $$ For example, if $n=2$, the matrix $$A = \pmatrix{0 & 1\cr 0 & 0\cr}$$ has $\varphi_v(A) = \overline{v_1} v_2$, and $|\varphi_v(A)| \le 1/2$ while $\|A\| = 1$. What is true is that if $A$ is a positive member of the $C^*$-algebra (corresponding to a positive-definite matrix) there is a state $\varphi$ with $\varphi(A) = \|A\|$.