Strong $L_p$-regularity under KMS symmetry

Determine whether strong $L_p$-regularity follows from KMS symmetry for primitive finite-dimensional quantum Markov semigroups, thereby establishing whether the comparison between the $L_2$-log-Sobolev constant and the modified log-Sobolev constant can be obtained without an additional regularity assumption.

Background

For primitive KMS-symmetric quantum Markov semigroups, the paper records the inequality chain $\alpha_H\leq\alpha_2\overset{(*)}{\leq}\alpha_1$, where the comparison marked $(*)$ requires strong $L_p$-regularity. This regularity condition is known for GNS-symmetric semigroups, but its relationship with KMS symmetry is unresolved.

The paper’s main results later remove the need for an $L_p$-regularity hypothesis by deriving modified log-Sobolev bounds from hyperboundedness and centered coercivity. Nevertheless, whether KMS symmetry alone implies strong $L_p$-regularity remains an explicitly identified unresolved question.