Bulk-boundary asymptotic equivalence of two strict deformation quantizations

The existence of a strict deformation quantization of X k = S(M k (C)), the state space of the k × k matrices M k (C) which is canonically a compact Poisson manifold (with stratified boundary), has recently been proved by both authors and Landsman (Rev Math Phys 32:2050031, 2020. https://doi.org/10.1142/S0129055X20500312). In fact, since increasing tensor powers of the k × k matrices M k (C) are known to give rise to a continuous bundle of C∗-algebras over I = {0} ∪ 1/N ⊂ [0, 1] with fibers A1/N = M k (C)⊗N and A0 = C(X k ), we were able to define a strict deformation quantization of X k à la Rieffel, specified by quantization maps Q1/N : / ̃A0 → A1/N , with ̃A0 a dense Poisson subalgebra of A0 . A similar result is known for the symplectic manifold S2 ⊂ R3 , for which in this case the fibers A′ 1/N = M N +1(C) ∼= B(SymN (C2)) and A′ 0 = C(S2) 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 : ̃A′ 0 → A′ 1/N . In this paper, we focus on the particular case X2 ∼= B3 (i.e., the unit three-ball in R3 ) and show that for any function f ∈ ̃A0 one has limN →∞ ||(Q1/N ( f ))|SymN (C2) − Q′ 1/N ( f |S2 )||N = 0, where SymN (C2) denotes the symmetric subspace of (C2)N ⊗. Finally, we give an application regarding the (quantum) Curie–Weiss model.