2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/94762For any finite dimensional C^*-algebra A with a trace vector \vec s whose entries are rational numbers, we give an endomorphism Φof the hyperfinite II_1 factor R such that: for all k \in \mathbb {N}, Φ^k (R)' \cap R= \otimes^k A. The canonical trace τon R extends the trace vector \vec s on A. Therefore the minimal projection is not necessarily equivalent to each other.Operator AlgebrasFunctional Analysis46L37; 47B47text