GNS dimension from density matrix rank ($\dim H_{\rho }=d\mathrm{rank}\rho$)

For the state on a C-star algebra $a\mapsto \mathrm{Tr}(\rho a)$ on $M_d(\mathbb{C})$, the GNS construction has the displayed Hilbert-space dimension. Here $\rho$ is a density matrix. Its null space for the GNS seminorm consists of $b$ with $b{\rho }^{1/2}=0$, because $\mathrm{Tr}(\rho b^{\ast }b)=\Vert b{\rho }^{1/2}{\Vert }_{\mathrm{HS}}^2$. Thus the GNS space is the space of matrices $b{\rho }^{1/2}$ with the Hilbert-Schmidt inner product. If $\rho$ has rank $r$, diagonalization identifies these matrices with $r$ arbitrary columns, giving dimension $dr$. Left multiplication is the representation and ${\rho }^{1/2}$ is its cyclic vector. In particular a pure rank-one matrix state gives the defining $d$-dimensional representation, while a full-rank state gives a $d^2$-dimensional space.