Injectivity of the representation map π: 𝔄_t → M(A)

Ascertain whether the non-degenerate *-homomorphism π: 𝔄_t → M(A), defined by π(x) = (π_n(x))_n where π_n ranges over all finite-dimensional *-representations used to build A, is injective.

Background

The algebra A is constructed as the direct sum of matrix algebras A_n associated with irreducible *-representations π_n of 𝔄_t. This yields a canonical *-homomorphism π into the multiplier algebra M(A).

Injectivity of π would ensure that the Hopf *-algebra 𝔄_t embeds faithfully into M(A), simplifying the transfer of structure (e.g., comultiplication) from 𝔄_t to A. The paper proceeds without assuming this, indicating the status is unresolved within the work.