Bulk-boundary asymptotic equivalence of two strict deformation quantizations

The existence of a strict deformation quantization of $X_k=S(M_k(\mathbb{C}))$, the state space of the $k\times k$ matrices $M_k(\mathbb{C})$ which is canonically a compact Poisson manifold (with stratified boundary) has recently been proven by both authors and K. Landsman \cite{LMV}. In fact, since increasing tensor powers of the $k\times k$ matrices $M_k(\mathbb{C})$ are known to give rise to a continuous bundle of $C^{\ast }$-algebras over $I=\{0\}\cup 1\mathrm{/}\mathbb{N}\subset [0,1]$ with fibers $A_{1\mathrm{/}N}=M_k(\mathbb{C}{)}^{\otimes N}$ and $A_0=C(X_k)$, we were able to define a strict deformation quantization of $X_k$ `{a} la Rieffel, specified by quantization maps $Q_{1\mathrm{/}N}:{\tilde{A}}_0\to A_{1\mathrm{/}N}$, with ${\tilde{A}}_0$ a dense Poisson subalgebra of $A_0$. A similar result is known for the symplectic manifold $S^2\subset {\mathbb{R}}^3$, for which in this case the fibers $A_{1\mathrm{/}N}^{\mathrm{\prime }}=M_{N+1}(\mathbb{C})\cong B({\text{Sym}}^N({\mathbb{C}}^2))$ and $A_0^{\mathrm{\prime }}=C(S^2)$ form a continuous bundle of $C^{\ast }$-algebras over the same base space $I$, and where quantization is specified by (a priori different) quantization maps $Q_{1\mathrm{/}N}^{\mathrm{\prime }}:{\tilde{A}}_0^{\mathrm{\prime }}\to A_{1\mathrm{/}N}^{\mathrm{\prime }}$. In this paper we focus on the particular case $X_2\cong B^3$ (i.e the unit three-ball in ${\mathbb{R}}^3$) and show that for any function $f\in {\tilde{A}}_0$ one has ${\lim }_{N\to \mathrm{\infty }}\mathrm{\mid }\mathrm{\mid }(Q_{1\mathrm{/}N}(f)){\mathrm{\mid }}_{{\text{Sym}}^N({\mathbb{C}}^2)}-Q_{1\mathrm{/}N}^{\mathrm{\prime }}(f{\mathrm{\mid }}_{{}_{S^2}})\mathrm{\mid }{\mathrm{\mid }}_N=0$, were ${\text{Sym}}^N({\mathbb{C}}^2)$ denotes the symmetric subspace of $({\mathbb{C}}^2{)}^{N\otimes }$. Finally, we give an application regarding the (quantum) Curie-Weiss model.