M ar 2 00 3 POSITIVE VOICULESCU - BROWN ENTROPY IN NONCOMMUTATIVE TORAL AUTOMORPHISMS

We show that the Voiculescu-Brown entropy of a noncommutative toral automorphism arising from a matrix S ∈ GL(d,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.