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Increasing tensor powers of the $k\times k$ matrices $M_k({\mathbb{C}})$ areknown to give rise to a continuous bundle of $C^*$-algebras over $I=\{0\}\cup1/\mathbb{N}\subset[0,1]$ with fibers $A_{1/N}=M_k({\mathbb{C}})^{\otimes N}$and $A_0=C(X_k)$, where $X_k=S(M_k({\mathbb{C}}))$, the state space of$M_k({\mathbb{C}})$, which is canonically a compact Poisson manifold (withstratified boundary). Our first result is the existence of a strict deformationquantization of $X_k$ \`{a} la Rieffel, defined by perfectly naturalquantization maps $Q_{1/N}: \tilde{A}_0\rightarrow A_{1/N}$ (where$\tilde{A}_0$ is an equally natural dense Poisson subalgebra of $A_0$). Weapply this quantization formalism to the Curie--Weiss model (an exemplaryquantum spin with long-range forces) in the parameter domain where its$\mathbb{Z}_2$ symmetry is spontaneously broken in the thermodynamic limit$N\raw\infty$. If this limit is taken with respect to the macroscopicobservables of the model (as opposed to the quasi-local observables), it yieldsa classical theory with phase space $X_2\cong B^3$ (i.e\ the unit three-ball in$\R^3$). Our quantization map then enables us to take the classical limit ofthe sequence of (unique) algebraic vector states induced by the ground stateeigenvectors $\Psi_N^{(0)}$ of this model as $N\raw\infty$, in which thesequence converges to a probability measure $\mu$ on the associated classicalphase space $X_2$. This measure is a symmetric convex sum of two Dirac measuresrelated by the underlying $\Z_2$-symmetry of the model, and as such theclassical limit exhibits spontaneous symmetry breaking, too. Our proof ofconvergence is heavily based on Perelomov-style coherent spin states and atsome stage it relies on (quite strong) numerical evidence. Hence the proof isnot completely analytic, but somewhat hybrid.

Strict deformation quantization of the state space of $M_k(\mathbb{C})$ with applications to the Curie-Weiss model