# Bulk-boundary asymptotic equivalence of two strict deformation quantizations

May 09, 2020

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
$C^*$-algebras over $I=\{0\}\cup 1/\mathbb{N}\subset[0,1]$ with fibers
$A_{1/N}=M_k({\mathbb{C}})^{\otimes N}$ 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 $Q_{1/N}: \tilde{A}_0\rightarrow A_{1/N}$, 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 $A'_{1/N}=M_{N+1}(\mathbb{C})\cong B(\text{Sym}^N(\mathbb{C}^2))$
and $A_0'=C(S^2)$ form a continuous bundle of $C^*$-algebras over the same base
space $I$, and where quantization is specified by (a priori different)
quantization maps $Q_{1/N}': \tilde{A}_0' \rightarrow A_{1/N}'$. 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
$\lim_{N\to\infty}||(Q_{1/N}(f))|_{\text{Sym}^N(\mathbb{C}^2)}-Q_{1/N}'(f|_{_{S^2}})||_N=0$,
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.

Keywords:

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