Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4619801 | Journal of Mathematical Analysis and Applications | 2009 | 10 Pages |
Abstract
Let Câ(E) be the Câ-algebra associated with a locally finite directed graph E and AE be the AF core of Câ(E). For the topological entropy ht(ΦE) (in the sense of Brown-Voiculescu) of the canonical completely positive map ΦE on the graph Câ-algebra, it is known that if E is finiteht(ΦE)=ht(ΦE|AE)=hb(E)=hl(E), where hb(E) (respectively, hl(E)) is the block (respectively, the loop) entropy of E. In case E is irreducible and infinite, hl(E)⩽ht(ΦE|AE)⩽hb(Et) is known recently, where Et is the graph E with the edges directed reversely. Then by monotonicity of entropy, hl(E)⩽ht(ΦE) is clear. In this paper we show that ht(ΦE)⩽hb(Et) holds for locally finite infinite graphs E. The AF core AE is known to be stably isomorphic to the graph Câ-algebra Câ(EÃcZ) of certain skew product EÃcZ and we also show that ht(ΦEÃcZ)=ht(ΦE|AE). Examples Ep (p>1) of irreducible graphs with ht(ΦEp)=logp are discussed.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
Ja A Jeong, Gi Hyun Park,