Finite-dimensional Dirac representation for the quotient seminorm L_q

Determine whether, for a finite-dimensional C*-algebra A and a unital C*-subalgebra B ⊆ A, there exists a Dirac-type operator D acting on a finite-dimensional Hilbert space representation π of A such that the quotient seminorm L_q(a) = inf_{b∈B} ||a − b|| satisfies L_q(a) = ||[D, π(a)]|| for all a ∈ A; if not, establish general obstructions to such finite-dimensional realizations.

Background

The paper constructs Dirac-type operators that represent quotient seminorms L_q via commutators, typically on infinite-dimensional Hilbert spaces (e.g., direct sums or function spaces over a projective/commutant set of projections). This yields strong Leibniz properties and links to noncommutative geometry.

The author raises the question of whether such representations can be achieved on finite-dimensional Hilbert spaces. Resolving this would clarify the minimal analytic machinery needed to realize L_q via spectral triples or Dirac-type constructions and impact computational and structural aspects of quantum Hamming metrics.