Topological Entropy for the Canonical Endomorphism of Cuntz-Krieger Algebras

Abstract

It is shown that Voiculescu's topological entropy for the canonical endomorphism of a simple Cuntz-Krieger algebra O A equals the logarithm of the spectral radius of A.

Topological entropy and AF subalgebras of graph 𝐶*-algebras

“…Using the same idea as in the proof of [4, Proposition 2.6] one can prove the following, which is stated in [2] without a proof in the case where {ω λ } is an increasing sequence. We provide a proof only for the reader's convenience.…”

“…We refer the reader to [2] and [4] for the following useful properties and their proofs. Let A be an exact C * -algebra and let Φ : A → A be a cp map.…”

“…For a finite graph E, the topological entropy ht(Φ E ) has been obtained as follows (see [2], [5], [19], or [11]). …”

“…The definition extends very well to automorphisms of exact C * -algebras (as done by Brown in [4]) due to the deep result by Kirchberg [13] that exact C * -algebras are nuclearly embeddable. But without effort one can define ht(Φ) even for a completely positive (cp) map on an exact C * -algebra as described in [2]. Since a C * -subalgebra of an exact C * -algebra is always exact, if Φ : A → A is a cp map on an exact C * -algebra A and B is a Φ-invariant C * -subalgebra of A, then ht(Φ| B ) can be defined and the monotonicity ht(Φ| B ) ≤ ht(Φ) holds.…”

“…A typical one is the canonical which is isomorphic to C(X A ) in such a way that the restriction Φ A | D A corresponds to the shift map σ X A on the (compact) shift space X A associated with the incidence matrix A. The topological entropy of Φ A is then computed (see [5,2,11,19]) as ht(Φ A ) = log r(A) (r(A) is the spectral radius of A). But log r(A) = h top (X A ) is a well-known fact, so that one can deduce by [8] …”