Topological Entropy for the Canonical Endomorphism of Cuntz-krieger Algebras

and φA is the endomorphism induced on C(XA) by the one-sided subshift of finite type σA (see [6]) defined by (σAx)k = xk+1, x = (xk)k∈N ∈ XA. Therefore φA can be regarded as a non-commutative generalization of the one-sided subshift of finite type associated with the matrix A and the computation of its dynamical entropies is of some interest (see [4, Page 691]). When A(i, i) = 1, i ∈ Σ, one gets the Cuntz algebra ON whereN is the cardinality of Σ (see [5]). In this case φN = ∑N j=1Sj · S ∗ j is a genuine endomorphism (i.e. φN (XY ) = φN (X)φN (Y ), X, Y ∈ ON ) which invariates the AF-part FN = ⊗∞ 1 MN (C) ofON and φN |FN coincides with the noncommutative Bernoulli shift φN (X) = 1 ⊗ X, X ∈ FN . Furthermore, φN |DN is the classical one-sided Bernoulli shift. D. Voiculescu has introduced in [9] a notion of topological entropy for noncommutative dynamical systems (A,α), where A is a unital nuclear C∗-algebra and α an automorphism (or endomorphism) of A which extends the classical commutative topological entropy. In the noncommutative framework partitions of unity are being replaced by ucp map ([4],[9]). As pointed out by N. Brown (see [1]), Voiculescu’s definition carries on, with slight modifications, to the larger class of (not necessarily unital) exact C∗-algebras.