Topological entropy and the AF core of a graph -algebra

Let C ∗ ( E ) be the C ∗ -algebra associated with a locally finite directed graph E and A E be the AF core of C ∗ ( E ) . For the topological entropy ht ( Φ E ) (in the sense of Brown–Voiculescu) of the canonical completely positive map Φ E on the graph C ∗ -algebra, it is known that if E is finite ht ( Φ E ) = ht ( Φ E | A E ) = h b ( E ) = h l ( E ) , where h b ( E ) (respectively, h l ( E ) ) is the block (respectively, the loop) entropy of E. In case E is irreducible and infinite, h l ( E ) ⩽ ht ( Φ E | A E ) ⩽ h b ( E t ) is known recently, where E t is the graph E with the edges directed reversely. Then by monotonicity of entropy, h l ( E ) ⩽ ht ( Φ E ) is clear. In this paper we show that ht ( Φ E ) ⩽ h b ( E t ) holds for locally finite infinite graphs E. The AF core A E is known to be stably isomorphic to the graph C ∗ -algebra C ∗ ( E × c Z ) of certain skew product E × c Z and we also show that ht ( Φ E × c Z ) = ht ( Φ E | A E ) . Examples E p ( p > 1 ) of irreducible graphs with ht ( Φ E p ) = log p are discussed.