Equivalence of two crystallization constructions for SU(n+1)

Prove that the crystallized C*-algebras obtained from Giri–Pal’s universal C*-algebra defined by q→0 relations among scaled generators and from Matassa–Yuncken’s operator-limit construction on a fixed Hilbert space are isomorphic for SU(n+1).

Background

The paper discusses two distinct crystallization frameworks: the construction in [6], which defines C0(SU(n+1)) via universal relations satisfied by appropriately scaled generators, and the construction in [16], which realizes crystal limits as operator algebras formed from q→0 limits in a fixed faithful representation.

Although the present work realizes its crystallized algebra as operators on the same Hilbert space as the Soibelman representation, thereby suggesting compatibility, the authors note that a proof of isomorphism between these two crystallized C*-algebras is currently lacking.