Morita equivalence for Cuntz–Pimsner algebras over equal-size matrix summands

Determine whether, for a finite direct sum of equal-size matrix algebras B\cong\bigoplus_{a=1}^d M_n and a completely positive map A:B\to B, the associated Cuntz–Pimsner algebra \mathcal O_{B\otimes_A B} is Morita equivalent to the Cuntz algebra \mathcal O_{r(A)} on r(A) generators.

Background

The paper proves that for a single-vertex finite quantum graph (M_n,tr,A), the local quantum Cuntz–Krieger algebra is Morita equivalent, and hence stably isomorphic, to the Cuntz algebra on r(A) generators, where r(A) is the Kraus rank of A. The authors then construct a unital copy of \mathcal O_n inside the local quantum Cuntz–Krieger algebra associated to a dephasing graph on B\cong\bigoplus_{a=1}d M_n.

The proposed problem asks whether the single-matrix-algebra Morita-equivalence result extends to finite direct sums of matrix algebras of a common size. A positive answer would imply that, for suitable finite quantum graphs whose adjacency map has no central summand in its kernel, the corresponding local quantum Cuntz–Krieger algebra is stably isomorphic to \mathcal O_{r(A)}; in particular, the dephasing case would yield stable isomorphism with \mathcal O_{dn}.

References

The unital embedding of $n$ inside $L(\oplus{a=1}d M_n,\psi,D)$ could provide insight to the isomorphism class of $L(\oplus_{a=1}d M_n,\psi,D)$ if a generalization of Theorem~\ref{thm:morita_equivalence}, as the following, were to hold: {\em Let $B\cong \oplus_{a=1}d M_n$. Given a cp map $A:B\to B$, is the associated Cuntz--Pimsner algebra $_{B\otimes_A B}$, as in , Morita equivalent to the Cuntz algebra on $r(A)$ generators?}

— Local quantum Cuntz--Krieger algebras of dephasing quantum graphs  (2608.23520 - Ismert, 24 Aug 2026) in Section “Cuntz algebras from dephasing quantum graphs,” immediately following Theorem 4.2 (the proposed generalization of Theorem 2.5)