Approximation entropies in crossed products with an application to free shifts

Entropy for automorphisms of free groups

“…Later he also proved in [13] that h φ (σ) = 0 with respect to the uniqueσ-invariant state φ of C * r (F ∞ ). We also see the same phenomenon for the topological entropy in [3] and [8], that is, ht(σ) = 0.…”

A Geometric Approach to Voiculescu-Brown Entropy

“…In fact the closed subspace spanned by the unitaries in C * r (F ∞ ) corresponding to the generators of F ∞ is completely isomorphic to the closed subspace spanned by the elements e 1i ⊕e i1 in the direct sum R⊕C of the row and column Hilbert spaces in B(ℓ 2 ) [14] (see also Section 8.3 of [28]). Actually, if we take the operator space structure of a matrix algebra into consideration then the geometric hint from above applies precisely and directly in the noncommutative case if we switch to the nuclear setting of the Cuntz algebra O ∞ (which can be viewed as an infinite reduced free product of Toeplitz algebras-see Chapter 1 of [38]) and consider the automorphism defined by shifting the index on the canonical isometries {s k } k∈Z [6], for then arithmetic dynamical growth and hence zero Voiculescu-Brown entropy, at least at the local level of the canonical isometries, is readily seen by combining the fact that the closed subspace spanned by {s k } k∈Z is canonically completely isometric to the column Hilbert space in B(ℓ 2 ) (see Section 1 of [29]) with a result of Pop and Smith that permits us to use general completely contractive linear maps in the definition of Voiculescu-Brown entropy [31] and an appeal to Wittstock's extension theorem which permits us to extend completely contractive linear maps into B(H) for any Hilbert space H (see [27]).…”

“…By viewing the free shift as a quantization of the compactified shift on Z it therefore seems natural to expect zero Voiculescu-Brown entropy in light of the above geometric and topological considerations. We will not delve here into a rigorous explanation for zero entropy, referring again to [11,6] for proofs.…”

Dual entropy in discrete groups with amenable actions

“…[CNT]) with respect to the canonical trace on C * r (G) (and in the W * -algebra L(G)). Also, the proof above together with the results of [BC,Section 2] can be used to show that dual entropy dominates the W * -entropy defined in [Vo,Section 3] in the case that G is an amenable group. Corollary 3.5 (Vo,Prop.…”