A Note on the Approximation Entropies of Certain Shifts

Voiculescu has proposed several routes to quantum entropies. Among them, the notion of the "approximation entropies" is a group of four entropies with similar definitions, based on two kinds of approximation s. The C*-cases are extensions of the classical topological entropy and the W*-cases are those of the measure-theoretic one. In this paper, we will focus on the approximation entropies and investigate the entropies of Powers' binary shifts with some condition and the Jones shifts. §1. Preliminaries Voiculescu [15] has introduced quantum entropies of automorphisms of operator algebras, called approximation entropies. In this section, we will review the definitions and fix the notations. Using two kinds of approximation for the W*-case and the C*-case, Voiculescu has defined four approximation entropies which have similar definitions to each other. As Voiculescu said, one may think of approximation entropies as "growth"-entr opies and the key concept is the "(5-rank". The four approximation entropies are defined in the same way, as a matter of form, except for the "(5-rank". So we will only state the definition by subalgebra approximation for the W*-case in detail, (see [15] for the other cases.) Let Ji be a hyperfinite von Neuman algebra with a faithful normal tracial state i and 3F(JC) be the set of the unital finite-dimensional C*-subalgebra s of M. By &f(Ji) we denote the set of finite subsets of Ji and by Aut(^) the automorphism group of Jt. For a normal faithful state cp on M, set AutpT, (p) = {a e AutpT) | cp ° a = cp}. For CD e &f(Jf) and X a Ji, we shall write