I am learning the GNS representation for C* algebras, and I would like to make a first step by applying the GNS construction on the simple algebra of the 2x2 matrices on the complex field. I would like to prove the two following things, assuming that the generic matrix of the algebra is written as A = \begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}
Assume firstly that the state is $\omega_1$ such that $\omega_1$ (A) = a. Compute the GNS representation, and show if the commutant of the resulting algebra is trivial.
Assume now that the state is $\omega_2$ such that $\omega_2$ (A) = Tr[$\rho$ A], where $\rho$ = \begin{pmatrix} 1/2 & 0 \\ 0 & 1/2 \\ \end{pmatrix} Compute the GNS representation, and show if the commutant of the resulting algebra is trivial.
I expect that in case 1 the state is pure, so the commutant should be trivial. However in case 2 the state is a mixed state, so the commutant should not be trivial. Can you please help me with that? Thank you in advance.
Update: I think in this post there is the solution: https://physics.stackexchange.com/questions/334051/algebraic-formalism-of-quantum-mechanics