Positive Voiculescu–Brown entropy in non-commutative toral automorphisms

We show that the Voiculescu–Brown entropy of a non-commutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$ is at least half the value of the topological entropy of the corresponding classical toral automorphism. We also obtain some information concerning the positivity of local Voiculescu–Brown entropy with respect to single unitaries. In particular we show that if $S$ has no roots of unity as eigenvalues then the local Voiculescu–Brown entropy with respect to every product of canonical unitaries is positive, and also that in the presence of completely positive CNT entropy the unital version of local Voiculescu–Brown entropy with respect to every non-scalar unitary is positive.