TOPOLOGICAL ENTROPY OF FREE PRODUCT AUTOMORPHISMS

N.P. BROWN, K. DYKEMA AND D. SHLYAKHTENKO

Abstract. Using free probability constructions involving the Cuntz–Pimsner C^{∗}–algebra we show that the topological entropy ht(α ∗ β) of the free product of two automorphisms is given by the maximum max(ht(α), ht(β)). As applications, we show in full generality that free shifts have topological entropy zero. We show that any separable nuclear C^{∗}–dynamical system can be covariantly embedded into O_{2} and O_{∞} in an entropy–preserving way. It follows that any nuclear simple purely infinite C^{∗}–algebra admits an automorphism with any given value of entropy. We also show that the free product of two automorphisms satisfies the Connes–Narnhofer–Thirring variational principle, if the two automorphisms do.

1. Introduction.

The study of non–commutative topological dynamical systems, in other words, automorphisms of C^{∗}–algebras, goes back a long way. Among the most important invariants of such systems is entropy, first introduced in the operator algebra context by Connes and Størmer in [13]. (See [38] for an extensive survey of this subject.) In this paper, we deal with the non–commutative generalization of topological entropy, discovered by Voiculescu [44] for nuclear C^{∗}–algebras and extended by Brown [7] to the exact case. Although this entropy has been computed in many examples, we are very far from a good understanding of its behavior. For example, to this day we do not know in full generality the exact value of the topological entropy ht(α ⊗ β) of a tensor product of two automorphisms. In general the inequality ht(α ⊗ β) ≤ ht(α) + ht(β) is known. Equality is only known to hold when both α and β satisfy a CNT–variational principle.

In this paper, we address four general types of questions regarding topological entropy. The first question concerns the behavior of topological entropy under free products. The

precise question is the value of ht(α ∗ β) of the topological entropy of the (possibly amalgamated) reduced free product automorphism α ∗ β. The main theorem of this paper states that ht(α ∗ β) = max(ht(α), ht(β)), if the free product is with amalgamation over a finite dimensional C^{∗}–algebra. One surprising feature of this result is that the answer is precise — this is to be contrasted with the situation for tensor products. Although the free product of C^{∗}–algebras is more complex than the tensor product, it seems to behave more like the direct sum for the purposes of entropy, which is, curiously, close to its behaviour for the purposes of K–theory. One direct consequence of our result is that the free shift on an arbitrary reduced free product ∗_{i∈ℤ}(A, ϕ) has zero entropy.

Date: October 10, 2000.

N.B. and D.S. partially supported by NSF postdoctoral fellowships. K.D. partially supported by NSF Grant No. DMS 0070558. N.B. is currently an MSRI Postdoctoral Fellow.

Giving a bound on the topological entropy of a free product of two automorphisms can be considered a refinement of the result [19] that the reduced amalgamated free product of two exact C^{∗}–algebras is exact. Indeed, exactness of a C^{∗}–algebra B is the statement that ht(id : B → B) < +∞. The reader will find that our proof is related to the argument given in [20] for the exactness of reduced amalgamated free products.

The second general question concerns the Connes–Narnhofer–Thirring (CNT) variational principle. The Connes–Narnhofer–Thirring (CNT) entropy [12] is a generalization to non–commutative measure spaces of the classical Kolmogorov–Sinai entropy. An automorphism α is said to satisfy the CNT variational principle if its topological entropy is equal to the supremum of the CNT entropy computed with respect to all invariant states. Although this principle fails for general non–commutative dynamical systems, we show that if two automorphisms α ∈ Aut(A) and β ∈ Aut(B) satisfy the CNT variational principle, then so does α ∗ β.

The third general question concerns embeddings of dynamical systems. Kirchberg has shown that any separable exact C^{∗}–algebra can be embedded into the Cuntz algebra on two generators. We show that any nuclear C^{∗}–dynamical system can be covariantly embedded into the Cuntz algebra O_{∞} in an entropy–preserving way.

The last question concerns the possible values of entropy that can be attained by automorphisms of a given C^{∗}–algebra A, i.e., the set T E(A) = ht(Aut(A)). This is clearly an invariant of A. For instance, T E(A) = {+∞} is A is not exact; T E(A) = {0} if A is finite dimensional. We show that T E(O_{∞}) = [0, +∞]. Thanks to Kirchberg’s absorption results, this implies that any separable purely infinite nuclear simple C^{∗} –algebra A admits an automorphism with any given value of entropy; i.e., T E(A) = [0, +∞].

The main result of this paper, computing entropy of amalgamated free products of automorphisms, applies only to the case that amalgamation takes place over a finite dimensional subalgebra. Our results are likely to extend to the case of amalgamation over an arbitrary C^{∗}–subalgebra, if the following question can be answered in the affirmative:

Question 1.1. Let 0 ↪ I → E →^{π} B → 0 be an exact sequence of C^{∗}–algebras, which is split (i.e., there exists a ∗–homomorphism η : B → E, so that π ∘ η = id_{B} ). Let α be an automorphism of E, so that α(I) = I; denote the resulting automorphism of B by α̂. Assume that η ∘ α̂ = α ∘ η. Is it true that ht(α) ≤ max(ht(α̂), ht(α|_{I}))?

Acknowledgement.An early part of this work was carried out at Institute Henri Poincaré, Paris, France, to which the authors would like to express their warmest gratitude for the exciting and encouraging atmosphere. We would like to thank also N. Ozawa and E. Germain for helpful conversations. K.D. would like to thank Uffe Haagerup and Odense University for their kind hospitality when part of this work was carried out.

2. Preliminaries.

2.1. A review of topological entropy. Let A be an exact C^{∗} algebra and let ω ⊂ A be a finite subset. Fix a faithful representation π : A → B(H). Define the set CP A(A) to be the set of triples (Φ, Ψ, X), where X is a finite dimensional C^{∗}–algebra, and Φ : A → X, Ψ : X → B(H) are contractive completely–positive maps. For ϵ > 0, define

rcp(ω, ϵ) = inf(rank(X) : (Φ, Ψ, X) ∈ CP A(A), ∀a ∈ ω ∥Ψ ∘ Φ(a) − π(a)∥ < ϵ).

(This quantity is independent of the choice of π, cf. [7]). If α ∈ Aut(A) is an automorphism, then its topological entropy ht(α) is defined as

sup_{ω⊂A} sup_{ϵ} lim sup_{n→∞} \frac{1}{n} log rcp(ω ∪ ⋯ ∪ α^{n−1}(ω), ϵ).

This definition, obtained in [7], gives the same quantity as Voiculescu’s original definition of dynamical topological entropy [44] for nuclear C^{∗}–algebras A. We summarize below some properties of ht(α), which we will need in this paper. The proofs can be found in [7].

Theorem. Let α ∈ Aut(A) be as above. Then 1. ht is monotone: if B ⊂ A, and α(B) ⊂ B, then ht(α|_{B}) ≤ ht(α).

2. If A = \overline{⋃ A_{n}}, with A_{n} ⊂ A_{n+1} subalgebras and α(A_{n}) ⊂ A_{n}, then ht(α) = lim_{n} ht(α|_{A_{n}}).

3. If β ∈ Aut(B), let α ⊗ β ∈ Aut(A ⊗_{min} B) be the tensor product automorphism. Then ht(α ⊗ β) ≤ ht(α) + ht(β). If A contains an α–invariant projection, then ht(α ⊗ β) ≥ ht(β).

4. If β ∈ Aut(B), let α ⊕ β ∈ Aut(A ⊕ B) be the direct sum automorphism. Then ht(α ⊕ β) = max(ht(α), ht(β)).

5. If β ∈ Aut(A) commutes with α, i.e., β ∘ α = α ∘ β, then α extends to the obvious automorphism ᾱ ∈ Aut(A ⋊_{β} ℤ). Then ht(ᾱ) = ht(α).

2.2. Amalgamated free products. Let D be a unital C^{∗}–algebra. Recall that a D–valued non–commutative probability space is a C^{∗}–algebra A, containing D as a unital subalgebra, and endowed with a conditional expectation E : A → D.

Let now A_{1} and A_{2} be two unital C^{∗}–algebras with a common unital subalgebra D and with conditional expectations E_{j} : A_{j} → D, so that the GNS representation associated to each E_{j} is faithful. Then there exists a C^{∗}–algebra A, generated by A_{1} and A_{2} (in such a way that the copies of D inside A_{j} ⊂ A are identified), a conditional expectation E : A → D, and such that A_{j} ⊂ A are free with respect to E (see [45]), and E gives rise to a faithful GNS representation of A. The C^{∗}–algebra A is the reduced amalgamated free product over D of (A_{1}, E_{1}) and (A_{2}, E_{2}) and is denoted

(A, E) = (A_{1}, E_{1}) ∗_{D} (A_{2}, E_{2}). (2.1)

When the conditional expectations E_{j} are clear from context we may write A = A_{1} ∗_{D} A_{2}. In the case that D = ℂ1 ⊂ A_{1}, A_{2}, then E_{1}, E_{2} and E are states and A is the reduced free product of A_{1} and A_{2}, denoted simply

(A, E) = (A_{1}, E_{1}) ∗ (A_{2}, E_{2}). (2.2)

Moreover, in the situation of a free product (2.1) or (2.2), one has conditional expectations Φ_{1} : A → A_{1} and Φ_{2} : A → A_{2} such that E_{j} ∘ Φ_{j} = E.

Definition 2.2.1. Let (A, E : A → D) be a D–valued non–commutative probability space. We say that α ∈ Aut(A) is an automorphism of this space, if (i) α(D) = D and (ii) α ∘ E = E ∘ α.

Notice that in the case that D = ℂ, an automorphism of a D–probability space A is simply an automorphism of A, fixing the state E : A → D = ℂ.

Assume now that (A_{1}, E : A_{1} → D) and (A_{2}, E_{2} : A_{2} → D) are D–probability spaces. Assume further that α_{j} ∈ Aut(A_{j}) are D–space automorphisms, and α_{1}|_{D} = α_{2}|_{D}. Then is a unique automorphism α_{1}∗α_{2} : A_{1}∗_{D}A_{2} → A_{1}∗_{D}A_{2}, satisfying (α_{1}∗α_{2})|_{A_{j}⊂A_{1}∗_{D}A_{2}} = α_{j}, j = 1, 2.

We will need the following theorem of Blanchard and Dykema.

Theorem. [4] Let D and D̃ be unital C^{∗}–algebras, let (A_{1}, E_{1}), (A_{2}, E_{2}) be D–probability spaces and let (Ã_{1}, Ẽ_{1}) and (Ã_{2}, Ẽ_{2}) be D̃–probability spaces. Assume that the GNS representations associated to E_{j} and Ẽ_{j} are faithful (j = 1, 2). Let

(A, E) = (A_{1}, E_{1}) ∗_{D} (A_{2}, E_{2})
(Ã, Ẽ) = (Ã_{1}, Ẽ_{1}) ∗D̃ (Ã_{2}, Ẽ_{2}).

Suppose that π_{D} : D → D̃ is a (not necessarily unital) injective ∗–homomorphism and that there are injective ∗–homomorphisms π_{1} : A_{1} → Ã_{1} and π_{2} : A_{2} → Ã_{2} such that π_{D} ∘ E_{j} = Ẽ_{j} ∘ π_{j}. Then there is an injective ∗–homomorphism π : A → Ã such that π|_{A_{j}} = π_{j} (j = 1, 2) and Ẽ ∘ π = E.

2.3. K–theory for reduced free products. Thanks to fundamental work of E. Germain on K–theory of free products , one has the following six–term exact sequence for free products of nuclear C^{∗}–algebras.

Theorem. [22], [21], [23] Let A and B be unital nuclear C^{∗}–algebras and ϕ ∈ S(A), ψ ∈ S(B) be states with faithful GNS representations. Let C = (A, ϕ)∗(B, ψ) and ι_{A} : A ↪ C and ι_{B} : B ↪ C denote the canonical inclusions. Then there is an exact sequence

ℤ ⟶^{1↦[1_{A}]⊕−[1_{B}]} K_{0}(A) ⊕ K_{0}(B) ⟶^{(ι_{A})_{∗}+(ι_{B})_{∗}} K_{0}(C)
↑  ↓
K_{1}(C) ⟵^{(ι_{A})_{∗}+(ι_{B})_{∗}} K_{1}(A) ⊕ K_{1}(B) ⟵  0.

Moreover, if both A and B satisfy the Universal Coefficient Theorem of Rosenberg and Schochet (see e.g. [3]) then so does C.

2.4. Cuntz–Pimsner algebras. Let A be a C^{∗}–algebra, and H be a Hilbert bimodule over A. Assume that H is full, i.e., ⟨H, H⟩_{A} is dense in A. Let

F (H) = A ⊕ ⨁_{n≥1} H^{⊗_{A}n},

and for ξ ∈ H, let

l(ξ) : F (H) → F (H),
l(ξ) · ξ_{1} ⊗ ⋯ ⊗ ξ_{n} = ξ ⊗ ξ_{1} ⊗ ⋯ ⊗ ξ_{n}.

The Cuntz–Pimsner C^{∗}–algebra E(H) (cf. [33]) is then defined as C^{∗}(l(ξ) : ξ ∈ H). It is not hard to see that

l(ξ)^{∗}l(ζ) = ⟨ξ, ζ⟩_{A},

and hence A = \overline{⟨H, H⟩} ⊂ E(H), acting on the left of F (H). The projection from F (H) onto A ⊂ F (H) gives rise to a conditional expectation E : E(H) → A. We summarize some of the properties of E(H) below; the proofs can be found in [33], [37] and [20].

Theorem. Let H and A be as above. Then

1. If K ⊂ H is a Hilbert A–subbimodule which is full, then E(K) is canonically isomorphic to C^{∗}(l(ξ) : F (H) → F (H) s.t. ξ ∈ K) ⊂ E(H).

2. Let H′ be another full Hilbert A–bimodule, then E(H ⊕ H′) = E(H) ∗_{A} E(H′), where the reduced free product is taken with respect to the canonical conditional expectations from E(H) (respectively, E(H′)) onto A.

3. Assume that ᾱ : H → H is a linear map, so that for some α ∈ Aut(A),

(2.3)  ᾱ(a_{1} · ξ · a_{2}) = α(a_{1}) · ᾱ(ξ) · α(a_{2}), a_{1}, a_{2} ∈ A, ξ ∈ H
(2.4)  ⟨ᾱ(ξ_{1}), ᾱ(ξ_{2})⟩_{A} = α(⟨ξ_{1}, ξ_{2}⟩_{A}), ξ_{1}, ξ_{2} ∈ H.

Then there is a unique automorphism E(ᾱ) of the A–probability space (E(H), E : E(H) → A), so that E(ᾱ)|_{A⊂E(H)} = α, and E(ᾱ)(l(ξ)) = l(ᾱ(ξ)). The automorphism E(ᾱ) is called the Bogoljubov automorphism associated to ᾱ.

2.5. Some examples of Bogoljubov automorphisms. In the course of proving the main theorem of the paper, we will encounter a particular class of Bogoljubov automorphisms of E(H), which we will presently describe. Let D be a C^{∗}–algebra and (A, E : A → D) be a D–probability space. Let

K^{o} = A ⊗ A

(algebraic tensor product), and endow K^{o} with the A–valued inner product given by

⟨a ⊗ b, a′ ⊗ b′⟩_{A} = b^{∗}E(a^{∗}a′)b′, a, a′, b, b′ ∈ A.

Denote by K_{D} the Hilbert A–bimodule obtained from K^{o} after separation and completion. Another description of K_{D} is as the internal tensor product K_{D} = L^{2}(A, E) ⊗_{D} A. Notice that any automorphism α of the D–probability space A extends to a linear map ᾱ : K_{D} → K_{D}, satisfying equations (2.3) and (2.4).

Theorem. [35] E(K_{D}) ≅ (A, E)∗_{D}(D⊗T , id ⊗ψ), where T denotes the Toeplitz algebra (the algebra generated by the unilateral shift on ℓ^{2}(ℕ)), and ψ : T → ℂ is the vacuum state (corresponding to the vector δ_{1} ∈ ℓ^{2}(ℕ)). Moreover, E(ᾱ) corresponds in this isomorphism to α ∗ ((α|_{D}) ⊗ id).

Let (A, E : A → D) be a D–probability space, and let α be an automorphism of this D– probability space. Assume that D is finite dimensional. Let ϕ be an α|_{D}–invariant faithful trace on D. Consider the Hilbert A, A–bimodule

H = L^{2}(A, ϕ ∘ E) ⊗_{ℂ} A,

together with the vector ξ = 1 ⊗ 1 ∈ H and the inner product

⟨a_{1} ⊗ a_{2}, b_{1} ⊗ b_{2}⟩ = ⟨a_{1}, b_{1}⟩_{L^{2}(A,ϕ∘E)}a_{2}^{∗}b_{2}.

We will henceforth also write ϕ to mean ϕ ∘ E. Let U : H → H be given by

U(x ⊗ a) = V (x) ⊗ α(a),

where V : L^{2}(A, ϕ) → L^{2}(A, ϕ) is the unitary induced by α : A → A. Then UaU^{∗} = α(a) where a and α(a) act on the left of H.

Lemma 2.5.1. There exists a vector ζ ∈ H, with the following properties:

1. dζ = ζd for all d ∈ D

2. Uζ = ζ

3. ⟨ζ, aζ⟩ = E(a) for all a ∈ A.

Proof. Let U(D) denote the unitary group of D, endowed with Haar measure μ. Let

ζ′ = ∫_{u∈U(D)} uξu^{∗}dμ(u).

For each w ∈ U(D), we get

wζ′ = w ∫_{u∈U(D)} uξu^{∗}dμ(u) = (∫_{uw∈U(D)} wuξuw^{∗}dμ(u)) w = ζ′w,

so dζ′ = ζ′d for all d ∈ D. Furthermore,

Uζ′ = ∫_{u∈U(D)} U(u ⊗ u^{∗})dμ(u) = ∫_{u∈U(D)} α(u) · 1 ⊗ α(u^{∗})dμ(u)
= ∫_{u∈U(D)} α(u)ξα(u^{∗})dμ(u) = ζ′.

Lastly, for a ∈ A, set

Φ(a) = ⟨ζ′, aζ′⟩ = ∫_{u,v∈U(D)}⟨u ⊗ u^{∗}, av ⊗ v^{∗}⟩dμ(u)dμ(v)
= ∫_{u,v} uϕ(u^{∗}av)v^{∗}dμ(u)dμ(v)
= ∫_{u,v} ϕ(vu^{∗}a)uv^{∗}dμ(u)dμ(v)
= ∫_{w=uv^{∗},v} ϕ(wa)w^{∗}dμ(w)dμ(v)
= ∫_{w} ϕ(wa)w^{∗}dμ(w).

Assume now that E(a) = 0, i.e., ϕ(da) = 0 for all d ∈ D. Then

Φ(a) = ∫_{w} ϕ(wa)w^{∗}dμ(w) = 0.

Since whenever v ∈ U(D),

Φ(v) = ∫_{w} ϕ(wv)w^{∗}dμ(w) = ∫_{u=wv} ϕ(u)vu^{∗}dμ(u) = vΦ(1),

and any d ∈ D is a linear combination of unitaries from U(D), we get that

Φ(d) = dΦ(1), ∀d ∈ D.

Since for any a ∈ A, a = (a − E(a)) + E(a) and E(a) ∈ d, we get that

Φ(a) = Φ(a − E(a)) + Φ(E(a)) = E(a)Φ(1).

We also have dΦ(1) = Φ(d) = ⟨ζ′, dζ′⟩ = ⟨ζ′, ζ′d⟩ = Φ(1)d. Hence Φ(1) is in the center of D; moreover, Φ(1) = ⟨ζ′, ζ′⟩ ≥ 0 and α(Φ(1)) = α(⟨ζ′, ζ′⟩) = ⟨Uζ′, Uζ′⟩ = ⟨ζ′, ζ′⟩ = Φ(1). We claim that Φ(1) is invertible. Writing D = ⊕M_{n_{k}} as a direct sum of matrix algebras, we find that Φ(1) = ∫_{u∈U(D)} ϕ(u)u^{∗}dμ(u) = ⊕Φ_{k}, where Φ_{k} = α_{k} · 1_{M_{n_{k}}} · ∫_{u∈U(n_{k})} Tr(u)u^{∗}dμ(u), and α_{k} ≥ 0 is related to the value of ϕ on the minimal projection of D corresponding to the k-th matrix summand in the direct sum decomposition of D. To show that Φ(1) is invertible, is it sufficient to show that c_{k} = ∫_{u∈U(k)} Tr(u)u^{∗}dμ(u) is a strictly positive scalar for all k ≥ 1. Repeating the argument above with D replaced by M_{k}, we find that c_{k} is in the center of M_{k} and is non–negative. Furthermore,

Tr(c_{k}) = ∫_{u∈U(k)} Tr(u)Tr(u^{∗})dμ(u) = ∫_{u∈U(k)} |Tr(u)|^{2}dμ(u) > 0,

since the subset of u ∈ M_{k} with Tr(u) ≠ 0 has non–zero measure.

Now let ζ = Φ(1)^{−1/2}ζ′. Then Uζ = UΦ(1)^{−1/2}U^{∗}Uζ′ = Φ(1)^{−1/2}ζ′ = ζ; for all d ∈ D, dζ = dΦ(1)^{−1/2}ζ′ = Φ(1)^{−1/2}dζ′ = Φ(1)^{−1/2}ζ′d = ζd; and for all a ∈ A,

⟨ζ, aζ⟩ = ⟨Φ(1)^{−1/2}ζ′, aΦ(1)^{−1/2}ζ′⟩ = ⟨ζ′, Φ(1)^{−1/2}aΦ(1)^{−1/2}ζ′⟩
= Φ(Φ(1)^{−1/2}aΦ(1)^{−1/2})
= Φ(1)^{−1/2}E(a)Φ(1)^{−1/2}Φ(1)
= E(a),

as desired.

Corollary 2.5.2. Let α be an automorphism of a D–probability space (A, E : A → D) with dim(D) < ∞. Assume that E : A → D gives rise to a faithful GNS representation. Let ϕ be an α–invariant trace on D. Consider the automorphism β = α∗id on B = (A, ϕ∘ E) ∗ T , where T is the Toeplitz algebra with its vacuum state. Consider the automorphism γ = α ∗ (α|_{D} ⊗ id) on C = (A, E) ∗_{D} (D ⊗ T ). Then there exists a covariant embedding of (C, γ) into (B, β).

Proof. Let H be as above. Consider the Cuntz–Pimsner C^{∗}–algebra E(H), generated by the operators l(h) : h ∈ H and A. Then B = E(H) = C^{∗}(A, l(h) : h ∈ H) = C^{∗}(A, l(ξ)). The automorphism β is identified with the Bogoljubov automorphism E(U): indeed we have β(a) = α(a), a ∈ A and E(U)(l(a ⊗ b)) = l(U(a ⊗ b)) = l(α(a) ⊗ α(b)) = α(a)l(ξ)α(b) = β(al(ξ)b). Choose ζ as in Lemma 2.5.1. Let L = l(ζ). Then L^{∗}aL = ⟨ζ, aζ⟩ = E(a). Let K = \overline{AζA} ⊂ H. It follows that C^{∗}(A, L) = C^{∗}(aLb : a, b ∈ A) = C^{∗}(l(h) : h ∈ K) = E(K). It is easily seen that the map aζb ↦ a ⊗ b defines an isomorphism of K with the A bimodule K_{D} defined in §2.5. Hence by the results of that section, C^{∗}(A, L) ≅ C. Moreover, since

E(U)(a) = α(a), a ∈ A
E(U)(L) = l(U(ζ)) = l(ζ) = L

we get that E(U)|_{C^{∗}(A,L)} = γ.

3. Toplogical entropy in certain extensions

The main result of this section gives an estimate of topological entropy in certain extensions. We begin with a lemma which gives a way of estimating the δ–rank of finite sets in an extension.

Consider a short exact sequence 0 → I → E →^{π} B → 0 where E is a unital exact C^{∗}–algebra and assume that there exists a unital completely positive splitting ρ : B → E (i.e. π ∘ ρ = id_{B}).

Lemma 3.1. Let I, E, B, π and ρ be as above. Given ϵ > 0 there exists a δ > 0 with the following property: if ω = {ρ(b_{1}) + x_{1}, …, ρ(b_{s}) + x_{s}} ⊂ E is a finite set containing the unit of E and such that each b_{i} ∈ B, x_{i} ∈ I and ∥b_{i}∥, ∥x_{i}∥ ≤ 1, 1 ≤ i ≤ s and if 0 ≤ e ≤ 1_{E} is an element of I such that ∥[e, ρ(b_{i})]∥, ∥x_{i} − ex_{i}∥, ∥x_{i} − x_{i}e∥ < δ for 1 ≤ i ≤ s then

rcp_{E}(ω, 30ϵ) ≤ rcp_{I}(e^{1/n}ωe^{1/n}, ϵ) + rcp_{B}(π(ω), ϵ),

for any n > 1/ϵ.

Proof. Fix ϵ > 0. By the lemma in [1, page 332] we can find a δ̃ > 0 such that for every pair of elements f, g in the unit ball of E such that f ≥ 0 we have the implication

∥[f, g]∥ < δ̃ ⇒ ∥[f^{1/2}, g]∥ < ϵ.

Let δ = min(δ̃, ϵ^{2}/4).

So assume that ω = {ρ(b_{1}) + x_{1}, …, ρ(b_{s}) + x_{s}} ⊂ E is a finite set such that each b_{i} ∈ B, x_{i} ∈ I and ∥b_{i}∥, ∥x_{i}∥ ≤ 1, 1 ≤ i ≤ s and 0 ≤ e ≤ 1_{E} is an element of I such that ∥[e, ρ(b_{i})]∥, ∥x_{i} − ex_{i}∥, ∥x_{i} − x_{i}e∥ < δ for 1 ≤ i ≤ s. By our choice of δ we then have ∥[e^{1/2}, ρ(b_{i})]∥, ∥[(1_{E} − e)^{1/2}, ρ(b_{i})]∥ < ϵ for all 1 ≤ i ≤ s.

We also claim that ∥[e^{1/2}, x_{i}]∥ < ϵ for all 1 ≤ i ≤ s. To see this we first note that since e ≤ e^{1/2} we have that 1_{E} − e ≥ 1_{E} − e^{1/2}. Thus

∥e^{1/2}x − x∥^{2} = ∥x^{∗}(1_{E} − e^{1/2})^{2}x∥ ≤ ∥x^{∗}(1_{E} − e^{1/2})x∥ ≤ ∥x^{∗}(1_{E} − e)x∥ ≤ ∥x − ex∥,

for all x ∈ I with ∥x∥ ≤ 1. Similarly, ∥xe^{1/2} − x∥ ≤ ∥x − xe∥ and hence

∥[e^{1/2}, x_{i}]∥ ≤ ∥e^{1/2}x_{i} − x_{i}∥ + ∥x_{i}e^{1/2} − x_{i}∥ ≤ ∥ex_{i} − x_{i}∥^{1/2} + ∥x_{i}e − x_{i}∥^{1/2} < ϵ.

Since 0 ≤ e ≤ 1_{E} some routine functional calculus shows that ∥e^{1/2+1/n} − e^{1/2}∥ ≤ 2/n and ∥e^{1+2/n} − e∥ ≤ 2/n. Combining these inequalities, the inequalities in the previous paragraphs and a standard interpolation argument we get

∥[e^{1/2+1/n}, ρ(b_{i})]∥ < 4/n + ϵ,
∥[e^{1/2+1/n}, x_{i}]∥ < 4/n + ϵ,
∥ρ(b_{i})e^{1+2/n} − ρ(b_{i})e∥ ≤ 2/n,
∥x_{i}e^{1+2/n} − x_{i}e∥ ≤ 2/n.

These four inequalities will be needed at the end of the proof.

Assume that E ⊂ B(H) and B ⊂ B(K). By Arveson’s extension theorem we may assume that ρ : B → E ⊂ B(H) is defined on all of B(K) (and takes values in B(H)). Now choose (ϕ̃_{1}, ϕ̃_{2}, D_{1}) ∈ CP A(B) such that ∥ϕ̃_{2} ∘ ϕ̃_{1}(b_{i}) − b_{i}∥ < ϵ, 1 ≤ i ≤ s, and rank(D_{1}) = rcp_{B}(π(ω), ϵ). Using the techniques in the proof of Proposition 1.4 in [7] we may replace the (not necessarily unital) maps ϕ̃_{1} and ϕ̃_{2} with unital maps ϕ_{1} : B → D_{1} and ϕ_{2} : D_{1} → B(K) such that ∥ϕ_{2} ∘ϕ_{1}(b_{i})−b_{i}∥ < 14ϵ. (Here we use the facts that 1_{B} ∈ π(ω) and ∥b_{i}∥ ≤ 1, 1 ≤ i ≤ s.)

Similarly, choose (ψ_{1}, ψ_{2}, D_{2}) ∈ CP A(I) such that

∥ψ_{2} ∘ ψ_{1}(e^{1/n}(b_{i} + x_{i})e^{1/n}) − e^{1/n}(b_{i} + x_{i})e^{1/n}∥ < ϵ,

1 ≤ i ≤ s, and rank(D_{2}) = rcp_{I}(e^{1/n}ωe^{1/n}, ϵ). By Arveson’s extension theorem we may assume that ψ_{1} is defined on all of B(H).

Define χ_{1} : E → D_{1} ⊕ D_{2} by

χ_{1}(y) = ϕ_{1}(π(y)) ⊕ ψ_{1}(e^{1/n}ye^{1/n}),

for all y ∈ E and χ_{2} : D_{1} ⊕ D_{2} → B(H) by

χ_{2}(S ⊕ T) = (1_{E} − e)^{1/2}ρ(ϕ_{2}(S))(1_{E} − e)^{1/2} + e^{1/2}ψ_{2}(T)e^{1/2},

for all S ∈ D_{1}, T ∈ D_{2}. Since we have arranged that ϕ_{2} is unital (and ρ is unital by assumption) we see that χ_{2}(1_{D_{1}} ⊕ 1_{D_{2}}) = 1_{E} − e + e^{1/2}ψ_{2}(1_{D_{2}})e^{1/2} = 1_{E} − (e^{1/2}(1_{E} − ψ_{2}(1_{D_{2}}))e^{1/2}). Since ψ_{2} is a contractive completely positive map, this shows that χ_{2}(1_{D_{1}} ⊕ 1_{D_{2}}) is a positive operator of norm less than or equal to one. Since it is clear that χ_{2} is a completely positive map this, in turn, implies that χ_{2} is also a contractive map (see [31, Proposition 3.5]).

Hence (χ_{1}, χ_{2}, D_{1} ⊕ D_{2}) ∈ CP A(E) and

rank(D_{1} ⊕ D_{2}) = rcp_{I}(e^{1/n}ωe^{1/n}, ϵ) + rcp_{B}({π(ω)}, ϵ).

Thus we only have to check that ∥χ_{2} ∘ χ_{1}(ρ(b_{i}) + x_{i}) − ρ(b_{i}) − x_{i}∥ < 31ϵ, for 1 ≤ i ≤ s. But, letting y = ρ(b_{i}) + x_{i} we have

∥χ_{2} ∘ χ_{1}(y) − y∥ = ∥(1_{E} − e)^{1/2}ρ(ϕ_{2}(ϕ_{1}(b_{i})))(1_{E} − e)^{1/2} +
e^{1/2}ψ_{2} ∘ ψ_{1}(e^{1/n}(ρ(b_{i}) + x_{i})e^{1/n})e^{1/2} − ρ(b_{i}) − x_{i}∥
≤ ∥(1_{E} − e)^{1/2}ρ(ϕ_{2}(ϕ_{1}(b_{i})))(1_{E} − e)^{1/2}
− (1_{E} − e)^{1/2}ρ(b_{i})(1_{E} − e)^{1/2}∥
+ ∥e^{1/2}ψ_{2} ∘ ψ_{1}(e^{1/n}(ρ(b_{i}) + x_{i})e^{1/n})e^{1/2}
− e^{1/2}(e^{1/n}(ρ(b_{i}) + x_{i})e^{1/n})e^{1/2}∥
+ ∥(1_{E} − e)^{1/2}ρ(b_{i})(1_{E} − e)^{1/2} + e^{\frac{n+2}{2n}} (ρ(b_{i}) + x_{i})e^{\frac{n+2}{2n}}
−ρ(b_{i}) − x_{i}∥
≤ 15ϵ+∥[(1_{E} − e)^{1/2}, ρ(b_{i})]∥+∥[e^{\frac{n+2}{2n}} , ρ(b_{i})]∥+∥[e^{\frac{n+2}{2n}} , x_{i}]∥
+∥ρ(b_{i})(1_{E} − e) + (ρ(b_{i}) + x_{i})e^{1+2/n} − ρ(b_{i}) − x_{i}∥

≤ 18ϵ + 8/n + ∥ρ(b_{i})e^{1+2/n} − ρ(b_{i})e∥
+∥x_{i}e^{1+2/n} − x_{i}e∥
≤ 18ϵ + 12/n.

Hence for any n > 1/ϵ we have the desired inequality.

Remark 3.2. The previous lemma is easily generalized to arbitrary extensions, though a precise formulation is somewhat awkward (and does not appear to be useful for entropy calculations). The idea is that if E is a unital exact C^{∗}–algebra then the quotient map π : E → B is always locally liftable (cf. [25, Prop. 7.2]). Hence the δ–rank of any finite subset of E can be estimated in terms of finite subsets of I and B.

Though it will not be needed in what follows it seems appropriate to point out the following application.

Proposition 3.3. With I, E, B, π and ρ as above, let α ∈ Aut(E) be an automorphism such that α(I) = I and let α̂ ∈ Aut(B) be the induced automorphism. Assume that ρ ∘ α̂ = α ∘ ρ and there exists an approximate unit {e_{λ}} ⊂ I such that α(e_{λ}) = e_{λ}, for all λ (which happens, for example, if there exists a strictly positive element h ∈ I such that α(h) = h). Then

ht(α) = max(ht(α|_{I}), ht(α̂)).

Proof. By [7, Prop. 2.10] it suffices to show the inequality ht(α) ≤ max(ht(α|_{I}), ht(α̂)).

Let ϵ > 0 be given and ω = {ρ(b_{1})+x_{1}, …, ρ(b_{s})+x_{s}} ⊂ E be any finite set containing the unit of E and such that ∥b_{i}∥, ∥x_{i}∥ ≤ 1, 1 ≤ i ≤ s. Let {e_{λ}} ⊂ I be an approximate unit such that α(e_{λ}) = e_{λ}, for all λ. Since we can manufacture a quasicentral approximate unit out of the convex hull of {e_{λ}} [1], we may further assume that {e_{λ}} is quasicentral in E (and still fixed by α).

Choose δ > 0 according to the previous lemma and take λ large enough that ∥[e_{λ}, ρ(b_{i})]∥, ∥x_{i} − e_{λ}x_{i}∥, ∥, x_{i} − x_{i}e_{λ}∥ < δ for 1 ≤ i ≤ s. Since α^{j}(e_{λ}) = e_{λ} for all j ∈ ℕ it is clear that ∥[e_{λ}, α^{j}(ρ(b_{i}))]∥, ∥α^{j}(x_{i}) − e_{λ}α^{j}(x_{i})∥, ∥α^{j}(x_{i}) − α^{j}(x_{i})e_{λ}∥ < δ for 1 ≤ i ≤ s and all j ∈ ℕ. Hence letting ω_{I} = e_{λ}^{1/k}ωe_{λ}^{1/k} ⊂ I, for some k > 1/ϵ, the previous lemma implies that rcp(ω ∪ … ∪ α^{n}(ω), 30ϵ) is bounded above by

rcp_{I}(ω_{I} ∪ … ∪ α^{n−1}(ω_{I}), ϵ) + rcp_{B}(π(ω) ∪ … ∪ α̂^{n−1}(π(ω)), ϵ)

which is bounded above by

2 max(rcp_{I}(ω_{I} ∪ … ∪ α^{n−1}(ω_{I}), ϵ), rcp_{B}(π(ω) ∪ … ∪ α̂^{n−1}(π(ω)), ϵ)).

This inequality implies the result.

The next lemma is inspired by section 5 in [43].

Lemma 3.4. Let F ⊂ B(H) be a finite set of self-adjoint contractive operators on a Hilbert space H. Let P be a projection in B(H), of rank ℓ < ∞. Then for any δ > 0, there exists a positive finite–rank contraction X ∈ B(H), so that:

1. XP = P X = P

2. ∥[X, T]∥ < δ for all T ∈ F

3. The rank of X is no bigger than ℓ · (|F | + 1)^{(2/δ)+1}.

Proof. Denote by K_{1} ⊂ H the range of P. Define recursively

K_{n} = span{K_{n−1} ∪ ⋃_{T ∈F} T K_{n−1}}.

Let q = |F | be the cardinality of F. Then K_{n} has dimension at most (q + 1) times the dimension of K_{n−1}, so that dim K_{n} ≤ ℓ · (q + 1)^{n}.

Let P_{n} be the orthogonal projection onto K_{n}; then P_{n} are clearly an increasing sequence, and P_{i}P = P P_{i} = P = P_{1} for all i. Let

X_{n} = \frac{1}{n}(P_{1} + ⋯ + P_{n}).

Then X_{n}P = P X_{n} = \frac{1}{n}(P P_{1} + ⋯ + P P_{n}) = P for all n. Note that the rank of X_{n} is the same as that of P_{n}, which is bounded by ℓ · (q + 1)^{n}

Set Q_{n} = P_{n} − P_{n−1}, Q_{1} = P_{1}. Since T K_{n} ⊂ K_{n+1} and thus T P_{n} = P_{n+1}T P_{n}, if m − n > 2 then we have Q_{m}T Q_{n} = Q_{m}T P_{n}Q_{n} = Q_{m}P_{n+1}T P_{n}Q_{n} = 0 for all T ∈ F. Since T is self-adjoint, also Q_{n}T Q_{m} = 0 if m−n > 2. Hence Q_{n}T Q_{m} = 0 if |n−m| > 2. Let Z be the orthocomplement of \overline{∪K_{n}} in H. Then in the decomposition H = ⨁ Q_{n}H⊕Z, each T ∈ F has the form

( \begin{matrix}Q_{1}T Q_{1} & Q_{1}T Q_{2} & 0 \\ Q_{2}T Q_{1} & Q_{2}T Q_{2} & Q_{2}T Q_{3} & 0 \\ 0 & Q_{3}T Q_{2} & Q_{3}T Q_{3} & Q_{3}T Q_{4} & ... \\ & 0 & ... & ... & ... \\ & & ... & ... & 0 \end{matrix} ) ,

i.e., it is a “block tri–diagonal” matrix. Hence using the convention Q_{0} = 0 we have the identities

Q_{n}T = ∑_{j=−1,0,1} Q_{n}T Q_{n+j},

T Q_{n} = ∑_{j=−1,0,1} Q_{n+j}T Q_{n}.

Now X_{n} = Q_{1} + (1 − \frac{1}{n})Q_{2} + ⋯ + \frac{1}{n}Q_{n} = ∑_{i=1}^{n}(\frac{n−i+1}{n})Q_{i} so that for any T ∈ F,

X_{n}T = ∑_{i=1}^{n}(\frac{n − i + 1}{n})Q_{i}T = ∑_{j=−1,0,1} ∑_{i=1}^{n}(\frac{n − i + 1}{n})Q_{i}T Q_{i+j}.

Similarly,

T X_{n} = ∑_{i=1}^{n}(\frac{n − i + 1}{n})T Q_{i} = ∑_{j=−1,0,1} ∑_{i=1}^{n}(\frac{n − i + 1}{n})Q_{i+j}T Q_{i}

= ∑_{j=−1,0,1} ∑_{i=1}^{n}(\frac{n − i + 1 − j}{n} )Q_{i}T Q_{i+j}

Hence

∥X_{n}T − T_{n}X∥ = ∥ ∑_{j=−1,0,1} ∑_{i=1}^{n}(\frac{(n − i + 1) − (n − i + 1 − j)}{n})Q_{i}T Q_{i+j}∥

= ∥ ∑_{j=−1,0,1} ∑_{i=1}^{n} \frac{j}{n}Q_{i}T Q_{i+j}∥

≤ ∑_{j=−1,0,1} \frac{|j|}{n}∥ ∑_{i} Q_{i}T Q_{i+j}∥

≤ ∑_{j=−1,1} \frac{|j|}{n} = \frac{2}{n}

The last inequality is due to the fact that for a fixed j, the operator ∑ Q_{i}T Q_{i+j} is a blockdiagonal operator, with the blocks having orthogonal ranges, so that ∥ ∑ Q_{i}T Q_{i+j}∥ ≤ max{∥Q_{i}T Q_{i+j}∥} ≤ 1.

Choose the smallest integer n with n > \frac{2}{δ}, and set X = X_{n}. Then ∥[X, T]∥ ≤ \frac{2}{n} < δ; by construction, XP = P X = P and the rank of X is bounded by ℓ(q + 1)^{n} ≤ ℓ(q + 1)^{\frac{2}{δ} +1}.

We finally come to the main result of this section.

Theorem 3.5. Let A, B ⊂ B(H) be unital exact C^{∗}–algebras such that B ∩ K(H) = {0} and C = B ⊗ 1 + K(H) ⊗ A ⊂ B(H ⊗ H). For any two unitaries V, W ∈ B(H) such that AdV (B) = B and AdW (A) = A, the unitary U = V ⊗ W has the property that AdU(C) = C and

ht(AdU|_{C}) ≤ max(ht(AdV |_{B}), ht(AdW |_{A})).

Proof. We have a short exact sequence

0 → K(H) ⊗ A → C →^{π} B → 0 (3.1)

with splitting ρ : B → C given by ρ(b) = b ⊗ 1.

Let k_{1}, …, k_{N} ∈ K(H), a_{1}, …, a_{N} ∈ A and b_{1}, …, b_{N} ∈ B be self-adjoint elements, each of norm at most 1, so that each k_{i} has finite rank. Let L be the sum of the ranks of k_{1}, …, k_{N}. Let

ω = {k_{i} ⊗ a_{i} + b_{i} ⊗ 1 : 1 ≤ i ≤ N}.

Fixing ϵ > 0 and a positive integer n, let

ω(n) = ω ∪ (Ad U)(ω) ∪ ⋯ ∪ (Ad U)^{n−1}(ω)
= {1} ∪ {V^{r}k_{i}V^{−r} ⊗ W^{r}a_{i}W^{−r} + V^{r}b_{i}V^{−r} ⊗ 1 : 1 ≤ i ≤ N, 0 ≤ r ≤ n − 1}.

Then the sum of the ranks of

k_{1}, …, k_{N}, …, V^{n−1}k_{1}V^{−(n−1)}, …, V^{n−1}k_{N}V^{−(n−1)}

is at most nL, so there exists a projection P of rank nL, so that P V^{r}k_{j}V^{−r} = V^{r}k_{j}V^{−r}P = V^{r}k_{j}V^{−r} for all j = 1, …, N and r = 0, …, n − 1.

Consider the collection

F = {b_{1}, …, b_{N}} ∪ ⋯ ∪ {V^{n−1}b_{1}V^{−(n−1)}, …, V^{n−1}b_{N}V^{−(n−1)}},

of at most nN self-adjoint operators on B(H). Let δ be as in Lemma 3.1 for the given value of ϵ and the short exact sequence (3.1). By applying Lemma 3.4 with these choices of δ, P, F, and ℓ = nL, we find a positive, finite–rank X ∈ K(H), so that:

1. XP = P X = P, hence XV^{r}k_{j}V^{−r} = XP V^{r}k_{j}V^{−r} = P V^{r}k_{j}V^{−r} = V^{r}k_{j}V^{−r} = V^{r}k_{j}V^{−r}X for all 1 ≤ j ≤ N and 0 ≤ r ≤ n − 1

2. ∥[X, V^{r}b_{j}V^{−r}]∥ ≤ δ for all 1 ≤ j ≤ N and 0 ≤ r ≤ n − 1

3. The rank of X is at most nL · (nN + 1)^{(2/δ)+1}. Let e = X ⊗ 1. Then e satisfies the hypotheses of Lemma 3.1. Indeed,

∥[e, V^{r}b_{i}V^{−r} ⊗ 1]∥ = ∥[X, V^{r}b_{i}V^{−r}]∥ ≤ δ, 1 ≤ i ≤ N, 0 ≤ r ≤ n − 1

and

eV^{r}k_{i}V^{−r} ⊗ W^{r}a_{i}W^{−r} − V^{r}k_{i}V^{−r} ⊗ W^{r}a_{i}W^{−r} =
= (XV^{r}k_{i}V^{−r} − V^{r}k_{i}V^{−r}) ⊗ W^{r}a_{i}W^{−r} = 0

if 1 ≤ i ≤ N, and 0 ≤ r ≤ n − 1, etc. Hence by Lemma 3.1, as long as m > ϵ^{−1} we have

rcp_{C}(ω(n), 30ϵ) ≤ rcp_{K(H)⊗A}(e^{1/m}ω(n)e^{1/m}, ϵ) + rcp_{B}(π(ω(n)), ϵ).

Note that

e^{1/m}ω(n)e^{1/m} ={X^{2/m} ⊗ 1} ∪

∪ \begin{matrix}\{X^{1/m}V^{r}k_{i}V^{−r}X^{1/m} ⊗ W^{r}a_{i}W^{−r}+ & X^{1/m}V^{r}b_{i}V^{−r}X^{1/m} ⊗ 1 \\ : 1 ≤ i ≤ N, 0 ≤ r ≤ n − 1\}. \end{matrix}

Hence setting

ω_{A}(n) = {1} ∪ {W^{r}a_{i}W^{−r} : 1 ≤ i ≤ N, 0 ≤ r ≤ n − 1}
ω_{B}(n) = {1} ∪ {V^{r}b_{i}V^{−r} : 1 ≤ i ≤ N, 0 ≤ r ≤ n − 1}

we have

rcp_{K(H)⊗A}(e^{1/n}ω(n)e^{1/n}, ϵ) ≤ rcp_{A}(ω_{A}(n), ϵ/2) · rank(X).

On the other hand,

rcp_{B}(π(ω(n)), ϵ) = rcp_{B}(ω_{B}(n), ϵ).

Since the rank of X is at most nL(nN + 1)^{(δ/2)+1} it follows that

rcp(ω(n), 30ϵ) ≤ rcp_{A}(ω_{A}(n), ϵ/2)nL(nN + 1)^{2/δ+1} + rcp_{B}(ω_{B}(n), ϵ)
≤ 2(nL + 1)(nN + 1)^{2/δ+1} max ( rcp_{A}(ω_{A}(n), ϵ/2), rcp_{B}(ω_{B}(n), ϵ))

so that

lim sup_{n→∞} \frac{1}{n} log rcp(ω(n), 30ϵ) ≤
≤ max ( lim sup_{n→∞} \frac{1}{n} log rcp(ω_{A}(n), ϵ/2), lim sup_{n→∞} \frac{1}{n} log rcp(ω_{B}(n), ϵ))
≤ max ( ht(Ad W |_{A}), ht(Ad V |_{B}))_{.}

Since C can be written as the closure of the linear span of elements of the form appearing in ω, the statement of the theorem follows.

We shall record the following corollary, which will be the basis for entropy computations in this paper.

Corollary 3.6. Let A be a C^{∗}–algebra, α ∈ Aut(A), π : A → B(H) be a faithful representation of A and U ∈ U(H) be such that π(α(a)) = Uπ(a)U^{∗}. Assume that π(A) ∩ K(H) = {0}. Let A_{0} = A, α_{0} = α, π_{0} = π, H_{0} = H and U_{0} = U. Recursively construct C^{∗}–algebras A_{n}, α_{n} ∈ Aut(A_{n}), π_{n} : A_{n} → B(H_{n}) and U_{n} ∈ U(H_{n}) by setting

H_{n} = H_{n−1} ⊗ H,
A_{n} = K(H_{n−1}) ⊗ π(A) + π_{n−1}(A_{n−1}) ⊗ Id_{H},
π_{n} = obvious representation on H_{n},
U_{n} = U_{n−1} ⊗ U,
α_{n} = Ad U_{n}.

View A_{n−1} ⊂ A_{n} as A_{n−1} ≅ π_{n−1}(A_{n}) ⊗ Id_{H}. Let

A_{∞} = \overline{∪_{n}A_{n},} α_{∞} = lim_{→} α_{n}.

Then:

(i) ht(α_{∞}) = ht(α).

(ii) If γ is an injective endomorphism of A_{∞}, so that γ ∘ α_{∞} = α_{∞} ∘ γ, denote by ᾱ the obvious extension of α_{∞} to A_{∞} ⋊_{γ} ℕ. Then ht(ᾱ) = ht(α).

Proof. Statement (ii) follows from statement (i) and the results of [20]. Hence it is sufficient to prove (i); for that one only needs to prove that ht(α_{n}) ≤ ht(α), in view of the behavior of entropy with respect to inductive limits. We now proceed by induction on n. Since α_{0} = α, the statement is true for n = 0. Applying Theorem 3.5 to A_{n} = K(H_{n−1}) ⊗ π(A) + π_{n−1}(A_{n−1}) ⊗ Id_{H} gives ht(α_{n}) ≤ max(ht(α_{n−1}), ht(α)), which is equal to ht(α) by the induction hypothesis.

4. Free products with the Toeplitz algebra.

The main technical result of this section (which will be used to prove the more general result about entropy of amalgamated free products of automorphisms) is the following theorem.

Theorem 4.1. Let α be an automorphism of a D–probability space (A, E : D → A). Assume that D is finite dimensional. Assume that the GNS representation associated to E is faithful. Let T ⊂ B(ℓ^{2}) be the Toeplitz algebra generated by the unilateral shift ℓ(δ_{n}) = δ_{n+1} (n ≥ 1), and ψ be the vector state on T associated to δ_{1} ∈ ℓ^{2}. Consider on the algebra (A, E) ∗_{D} (D ⊗ T , id_{D} ⊗ψ) the automorphism α ∗ (α|_{D} ⊗ id). Then ht(α ∗ (α|_{D} ⊗ id)) = ht(α).

Because of Corollary 2.5.2 and monotonicity of ht, it is sufficient to prove Theorem 4.1 in the particular case that D = ℂ. For convenience, we’ll restate this particular case as

Proposition 4.2. Let A be a unital C^{∗}–algebra, ϕ : A → ℂ a state with a faithful GNS representation. Let α ∈ Aut(A) be an automorphism, so that ϕ ∘ α = ϕ . Consider the algebra (A, ϕ) ∗ (T , ψ). Then ht(α ∗ id) = ht(α).

Proof. Let H = L^{2}(A, ϕ) be the GNS Hilbert space associated to A, and let 1 ∈ H be the cyclic vector associated to ϕ. Let O_{2} be the Cuntz algebra on two generators [14]. Without loss of generality, by replacing A with A ⊗ O_{2} and α with α ⊗ id we may assume that the GNS representation π : A → B(H) satisfies π(A) ∩ K(H) = {0}. Let U : H → H be the unitary induced on H = L^{2}(A, ϕ) by α. We shall covariantly identify ((A, ϕ)∗(T , ψ), α∗id) as the crossed product by an certain endomorphism γ of the algebra A_{∞} described in Corollary 3.6, taken with the automorphism ᾱ. By Corollary 3.6, we then have ht(α) = ht(ᾱ), which, in view of our identification, is the same as ht(α ∗ id), hence proving the Proposition. The remainder of the proof is essentially a special case of the techniques used in [20] where more general Cuntz–Pimsner algebras were shown to have a crossed product structure.

Consider now the Hilbert space

F = L^{2}(A, ϕ) ⊕ ⨁_{n≥2} L^{2}(A, ϕ)^{⊗n},

and the representation ρ : A → B(H) given by ρ = π ⊕ (⨁_{n≥2} π ⊗ 1). Consider the isometry ℓ : F → F defined by

ℓ(ξ) = 1 ⊗ ξ,

where 1 denotes the image of the unit of A in L^{2}(A, ϕ), and ξ ∈ F. Denote by U : L^{2}(A, ϕ) → L^{2}(A, ϕ) the unitary implementing α. Denote by V the unitary U ⊕ ⨁_{n} U^{⊗n} acting on the Hilbert space F. One sees that Ad_{V} (ℓ) = ℓ, and Ad_{V} (ρ(a)) = ρ(α(a)).

Claim 4.3. (C^{∗}(ρ(A), ℓ), Ad_{V}) ≅ ((A, ϕ) ∗ (T , ψ), α ∗ id).

Proof. This actually follows from §2.5, since C^{∗}(ρ(A), ℓ) is isomorphic to the Cuntz– Pimsner algebra associated to the A, A bimodule L^{2}(A, ϕ) ⊗ A. For the reader’s convenience, we give a proof.

Let θ be the vector–state on B(F), associated to the vector 1 ∈ L^{2}(A, ϕ) ⊕ 0 ⊂ F. Then ℓ^{∗}1 = 0, and one can easily verify that: (i) θ(a_{0}ℓ ⋯ ℓa_{n}ℓ^{∗}a_{n+1} ⋯ ℓ^{∗}a_{n+m}) = 0 for all a_{j} ∈ ρ(A), n, m ≥ 0, n + m > 0; and (ii) ℓ^{∗}ρ(a)ℓ = ϕ(a), for all a ∈ A. It follows from [35] that A and C^{∗}(ℓ) are free in (B(F), θ). Since C^{∗}(ℓ) is (obviously) isomorphic to T , θ|_{C^{∗}(ℓ)} = ψ, and since ρ is injective and θ|_{ρ(A)} = ϕ, the claim is proved.

From now on, write C = C^{∗}(A, ℓ).

Denote by C_{n} the closed linear span

C_{n} = \overline{span}\{a_{0}ℓa_{1} ⋯ ℓa_{m}ℓ^{∗}a_{m+1}ℓ^{∗} ⋯ a_{2m} : m ≤ n, a_{0}, …, a_{2m} ∈ ρ(A)\}.

(each monomial above has exactly m terms equal to ℓ and the same number of terms equal to ℓ^{∗}). Note that because of the relation ℓ^{∗}ρ(a)ℓ = ϕ(a), a ∈ A, each C_{n} is a C^{∗}–subalgebra of C. The action of w = a_{0}ℓa_{1} ⋯ ℓa_{n}ℓ^{∗}a_{n+1}ℓ^{∗} ⋯ a_{2n} ∈ C_{k} on a vector ξ = ξ_{1} ⊗⋯⊗ξ_{r} ∈ H can be described as follows:

w · ξ = 0,  if n ≥ r
w · ξ = (a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ (a_{n} · ξ_{n+1}) ⊗ ⋯ ⊗ ξ_{r}) ∏_{j=1}^{n}⟨ξ_{j}, a_{2n−j+1}^{∗}⟩, otherwise

(here we identify a_{j} ∈ ρ(A) with a_{j} · 1 ∈ L^{2}(A, ϕ) ⊂ H).

Denote by C_{∞} the C^{∗}–algebra \overline{∪_{n≥0}C_{n}}. Then ℓC_{n}ℓ^{∗} ⊂ C_{n+1} and hence γ = ℓ · ℓ^{∗} determines an endomorphism of C_{∞}. It can be easily seen that C is isomorphic to the crossed product of C_{∞} by this endomorphism (cf. the discussion after Proposition 1.2 in [20] for the definition). Moreover, Ad_{V} leaves C_{∞} invariant and commutes with the endomorphism ℓ · ℓ^{∗} ([20, Proposition 1.5]).

It remains to show that (C_{n}, Ad_{V} |_{C_{n}}) ≅ (A_{n}, α_{n}), where A_{n} and α_{n} are as in Corollary 3.6.

We proceed by induction. In the case that n = 0, C_{0} = A, Ad_{V} |_{C_{0}} = α.

Let

F_{n} = L^{2}(A, ϕ) ⊕ ⨁_{k=2}^{n+1} L^{2}(A, ϕ)^{⊗k} ⊂ F.

Then F_{n} is invariant under the action of C_{n} on F. Consider the isomorphism W

W : L^{2}(A, ϕ) ⊕ ⨁_{k≥2} L^{2}(A, ϕ)^{⊗k} →

(L^{2}(A, ϕ) ⊕ ⨁_{k=2}^{n+1} L^{2}(A, ϕ)^{⊗k}) ⊗ (ℂ ⊕ ⨁_{m≥1} L^{2}(A, ϕ)^{⊗m(n+1)}) = F_{n} ⊗ K_{n}

For each a ∈ C_{n}, W^{∗}aW : F_{n} ⊗ K_{n} → F_{n} ⊗ K_{n} has the form a|_{F_{n}} ⊗ Id_{K_{n}}. It follows that the representation of C_{n} on F_{n} is faithful.

Denote by B_{n} ⊂ C_{n} the ideal

B_{n} = span{a_{0}ℓa_{1} ⋯ ℓa_{n}ℓ^{∗}a_{n+1}ℓ^{∗} ⋯ a_{2n} : a_{0}, …, a_{2n} ∈ ρ(A)}.

Write

F_{n} = L^{2}(A, ϕ) ⊕ ⨁_{k=2}^{n+1} L^{2}(A, ϕ)^{⊗k} = F_{n−1} ⊕ L^{2}(A, ϕ)^{⊗(n+1)}.

Then B_{n} · F_{n−1} = 0; hence the representation of B_{n} obtained by restricting its action to the space L^{2}(A, ϕ)^{⊗(n+1)} is faithful. The action of a_{0}ℓa_{1} ⋯ ℓa_{n}ℓ^{∗}a_{n+1}ℓ^{∗} ⋯ a_{2n} ∈ B_{n} on ξ_{1} ⊗ ⋯ ⊗ ξ_{n+1} ∈ L^{2}(A, ϕ)^{⊗(n+1)} can be explicitly written as

a_{0}ℓa_{1} ⋯ ℓa_{n}ℓ^{∗}a_{n+1}ℓ^{∗} ⋯ a_{2n} · ξ_{1} ⊗ ⋯ ⊗ ξ_{n} ⊗ ξ_{n+1} =
⟨ξ_{1}, a_{2n}^{∗}⟩ ⋯ ⟨ξ_{n}, a_{n+1}^{∗}⟩a_{0} ⊗ ⋯ ⊗ a_{n−1} ⊗ (a_{n+1} · ξ_{n+1}).

Denote by θ(a_{0}, …, a_{n−1}, a_{n+1}, …, a_{2n}) ∈ B(L^{2}(A, ϕ)^{⊗n}) the compact operator given by

θ(a_{0}, …, a_{n−1}^{∗}, a_{n+1}, …, a_{2n})ξ_{1} ⊗ ⋯ ⊗ ξ_{n} = ⟨ξ_{1}, a_{2n}^{∗}⟩ ⋯ ⟨ξ_{n}, a_{n+1}^{∗}⟩a_{0} ⊗ ⋯ ⊗ a_{n−1}.

Then the map

a_{0}ℓa_{1} ⋯ ℓa_{n}ℓ^{∗}a_{n+1}ℓ^{∗} ⋯ a_{2n} ↦ θ(a_{0}, …, a_{n−1}^{∗}, a_{n+1}, …, a_{2n}) ⊗ a_{n}
∈ K(L^{2}(A, ϕ)^{⊗n}) ⊗ A ⊂ B(L^{2}(A, ϕ)^{⊗n} ⊗ L^{2}(A, ϕ))

is a C^{∗}–algebra isomorphism of B_{n} with K(L^{2}(A, ϕ)^{⊗n}) ⊗ A.

For all m ≤ n + 1, the subspaces L^{2}(A, ϕ)^{⊗m} ⊂ F_{n} are invariant under the action of C_{n}. Denote by ρ_{n} the representation of C_{n} obtained by restricting its action on H to the space L^{2}(A, ϕ)^{⊗(n+1)}. The image ρ_{n}(A_{n−1}) lies inside B(L^{2}(A, ϕ)^{⊗(n−1)}) ⊗ A ⊗ Id_{L^{2}(A,ϕ)}. By our assumption, π(A) ∩ K(L^{2}(A, ϕ)) = {0}. This implies that ρ_{n}(C_{n}) ∩ ρ_{n}(B_{n}) = ρ_{n}(C_{n}) ∩ K(L^{2}(A, ϕ)^{⊗n}) ⊗ A = {0}.

We claim that ρ_{n} is faithful. Indeed, assume it’s not, and some 0 ≠ a ∈ C_{n} is annihilated by ρ. Write a = a_{n−1} + b, where a_{n−1} ∈ C_{n−1} and b ∈ B. Since ρ_{n}(a) = ρ_{n}(a_{n−1}) + ρ_{n}(b) = 0, and ρ_{n}(B_{n}) ∩ ρ_{n}(C_{n−1}) = {0}, it follows that ρ_{n}(a_{n−1}) = 0 and ρ_{n}(b) = 0. But for a_{n−1} ∈ C_{n−1}, ρ(a_{n−1}) = ρ_{n−1}(a_{n−1}) ⊗ Id_{L^{2}(A,ϕ)}. Proceeding inductively, we see that ρ_{n} is faithful.

We have therefore proved that

C_{n} = ρ_{n−1}(A_{n−1}) ⊗ Id_{L^{2}(A,ϕ)} +K(L^{2}(A, ϕ)^{n−1}) ⊗ π(A),

which means that C_{n} ≅ A_{n}. Since Ad V_{n}|_{C_{n}} is given by Ad_{U^{⊗n−1}⊗U}, it follows that the dynamical system (C_{n}, Ad V |_{C_{n}}) ≅ (A_{n}, α_{n}).

Corollary 4.4. Let α be an automorphism of a non–commutative probability space (A, E : A → D), and assume that D is finite dimensional. Assume that the GNS representation with respect to E is faithful. Let ᾱ be the Bogoljubov automorphism of the Cuntz–Pimsner C^{∗}–algebra described in §2.5. Then ht(ᾱ) = ht(α).

It would be interesting to find a formula for the entropy of more general Bogoljubov automorphisms of Cuntz–Pimsner algebras. For example, we believe that the Corollary above should hold for more general D.

5. Entropy for free products of automorphisms.

We have now almost arrived at the main result of the paper (Theorem 5.7), calculating entropy of the free product of two automorphisms. First, we shall prove several technical lemmas which reduce the general case to the case of a free product with the Toeplitz algebra as in Theorem 4.1.

Lemma 5.1. Let D ⊂ B ⊂ B C^{∗}–subalgebras, and let U ∈ B be a unitary. Let E be a conditional expectation from B onto D. Assume that: (i) C^{∗}(D, U) and B are free with respect to E and that (ii) [U, D] = 0 and E(U) = E(U^{∗}) = 0. Then the algebras B and UBU^{∗} are free with respect to E. If in addition, E(U^{k}) = 0 for all k ≠ 0, then the algebras {U^{k}BU^{−k}}_{k∈ℤ} are free with respect to E.

Proof. Since U is free from B and commutes with D, we see that for all b ∈ B, E(UbU^{∗}) = E(U(b − E(b))U^{∗}) + E(UE(b)U^{∗}) = 0 + E(E(b)UU^{∗}) = E(b). Now let b_{j} ∈ B be such that E(b_{j}) = 0. Then

E(b_{0}Ub_{1}U^{∗}b_{2}Ub_{3}U^{∗} ⋯) = 0

because U and B are free with respect to E. But then B and UBU^{∗} are free with respect to E, since for all j, Ub_{2j+1}U^{∗} ∈ UBU^{∗} ∩ ker E, and any element in UBU^{∗} ∩ ker E has the form UbU^{∗}, for some b ∈ B ∩ ker E.

The proof that {U^{k}BU^{−k}}_{k} are free with respect to E proceeds along similar lines.

The following lemma is included for completeness. We will use the implications (i) ⇒ (ii) ⇒ (iii) in the sequel.

Lemma 5.2. Let D ⊆ B ⊆ A be unital inclusions of unital C^{∗}–algebras, and suppose there are conditional expectations E_{B}^{A} : A → B and E_{D}^{B} : B → D. Let E_{D}^{A} = E_{D}^{B} ∘ E_{B}^{A} : A → D. Consider the following statements:

(i) The GNS representations associated to E_{B}^{A} and E_{D}^{B} are faithful. (ii) The GNS representation associated to E_{D}^{A} is faithful. (iii) The GNS representation associated to E_{B}^{A} is faithful. Then we have (i)⇒(ii)⇒(iii). None of the reverse implications hold in general.

Proof. We will denote by a → â the defining map from A onto a dense subspace of L^{2}(A, E_{D}^{A}), and similarly for the other L^{2} spaces. We will also need the isomorphism π : L^{2}(A, E_{D}^{A}) → L^{2}(A, E_{B}^{A}) ⊗_{B} L^{2}(B, E_{D}^{B}) given by π(â) = â ⊗ ˆ1.

(i)⇒(ii): Let a ∈ A. Choose a_{1} ∈ A so that (aa_{1})ˆ ∈ L^{2}(A, E_{B}^{A})\{0}, i.e. E_{B}^{A}(a_{1}^{∗}a^{∗}aa_{1}) ≠ 0. There exists and element b_{1} ∈ B such that

(E_{B}^{A}(a_{1}^{∗}a^{∗}aa_{1}))^{1/2}b_{1} ∈ L^{2}(B, E_{D}^{B})\{0},

i.e.

0 ≠ E_{D}^{B}(b_{1}^{∗}E_{B}^{A}(a_{1}^{∗}a^{∗}aa_{1})b_{1}) = E_{D}^{A}(b_{1}^{∗}a_{1}^{∗}a^{∗}aa_{1}b_{1}).

Thus (aa_{1}b_{1})ˆ ∈ L^{2}(A, E_{D}^{A})\{0}.

(ii)⇒(iii): Given a ∈ A take a_{1} ∈ A such that E_{D}^{A}(a_{1}^{∗}a^{∗}aa_{1}) ≠ 0, hence E_{B}^{A}(a_{1}^{∗}a^{∗}aa_{1}) ≠ 0.

A counter example to (ii)⇒(i) is provided by taking D = ℂ, B = ℂ ⊕ ℂ, A = M_{2}(ℂ), E_{B}^{A}((c_{ij})_{1≤i,j≤2}) = c_{11} ⊕ c_{22} and E_{D}^{B}(z_{1} ⊕ z_{2}) = z_{1}.

A counter example to (iii)⇒(ii) is provided by taking D, B and E_{D}^{B} as in the above example, A = M_{2}(ℂ) ⊕ M_{2}(ℂ) and E_{B}^{A}((c_{ij})_{1≤i,j≤2} ⊕ (d_{ij})_{1≤i,j≤2}) = c_{11} ⊕ d_{11}.

Lemma 5.3. Let E_{i} : A_{i} → D, i = 1, 2 be D–probability spaces, with automorphisms α_{i}. Assume that the GNS representations associated to E_{i} are faithful, and that α_{1}|_{D} = α_{2}|_{D}. Let E_{1} ⊕ E_{2} : A_{1} ⊕ A_{2} → D ⊕ D be the obvious conditional expectation. Consider the algebra M_{2}(D) of 2×2 matrices over D, and view D ⊕D ⊂ M_{2}(D) as diagonal matrices. Let F : M_{2}(D) → D ⊕ D be the conditional expectation F ( \begin{matrix}a & b \\ c & d \end{matrix}) = a ⊕ d.

Then there exists an isomorphism

ϕ : (A_{1} ⊕ A_{2}, E_{1} ⊕ E_{2}) ∗_{D⊕D} (M_{2}(D), F) ≅ ((A_{1}, E_{1}) ∗_{D} (A_{2}, E_{2})) ⊗ M_{2}(ℂ),

so that ϕ intertwines the automorphisms (α_{1} ⊕ α_{2}) ∗ (id ⊗α_{1}|_{D}) and (α_{1} ∗ α_{2}) ⊗ id.

Proof. Consider in M_{2}(D) ⊆ (A_{1} ⊕A_{2}, E_{1} ⊕E_{2})∗_{D⊕D} (M_{2}(D), F) the unitary w = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ). Consider the subalgebras

A′_{1} = a ⊕ 0,  a ∈ A_{1}
A′_{2} = w(0 ⊕ a)w^{∗},  a ∈ A_{2}.

Since w is free from A_{1} ⊕ A_{2} with amalgamation over D ⊕ D, it is easily seen from the definition of freeness and Lemma 5.1 that the algebras A′_{1} and A′_{2} are also free with amalgamation over D ⊕ 0. Furthermore, the restriction of (E_{1} ⊕ E_{2}) ∗ F to A′_{i} is E_{i}, and hence the GNS representation associated to each E_{i} is faithful. It follows from the embedding result (see §2.2) that A′_{1} and A′_{2} together generate the reduced free product A_{1} ∗_{D} A_{2}. It is easily seen that the algebra (A_{1} ⊕ A_{2}) ∗_{D⊕D} (M_{2}(D)) is generated by A′_{1}, A′_{2} and C^{∗}(1_{D} ⊕ 1_{D}, w) ≅ M_{2}(ℂ), and is isomorphic to (A_{1} ∗_{D} A_{2}) ⊗ M_{2}(ℂ) via the map

ϕ : A′_{1} ∋ a_{1} ⊕ 0 ↦ (\begin{matrix}a_{1} & 0 \\ 0 & 0 \end{matrix}) ∈ A_{1} ⊗ M_{2}(ℂ), a_{1} ∈ A_{1}
ϕ : A′_{2} ∋ w0 ⊕ a_{2}w^{∗} ↦ (\begin{matrix}a_{2} & 0 \\ 0 & 0 \end{matrix}) ∈ A_{2} ⊗ M_{2}(ℂ), a_{2} ∈ A_{2}
ϕ(w) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

It is clear that ϕ intertwines the automorphisms (α_{1} ⊕ α_{2}) ∗ id and (α_{1} ∗ α_{2}) ⊗ 1.

In the following lemma, O_{2} will be the Cuntz algebra [14] generated by isometries S_{1} and S_{2} such that S_{1}S_{1}^{∗} + S_{2}S_{2}^{∗} = 1 and σ will denote the state on O_{2} satisfying

σ(S_{i_{1}} ⋯ S_{i_{p}}S_{j_{q}}^{∗} ⋯ S_{j_{1}}^{∗}) = \begin{cases}2^{−p} & if p = q and i_{1} = j_{1}, ⋯, i_{p} = j_{p} \\ 0 & otherwise. \end{cases}

Note that σ is faithful.

Lemma 5.4. Let σ be the state on O_{2} as above. Let C be a commutative C^{∗}–algebra having a faithful state ρ and a unitary u ∈ C such that ρ(u) = 0. Consider the algebra B = ((ℂ ⊕ ℂ) ⊗ O_{2}, θ ⊗ σ) ∗ (C, ρ), where θ(a ⊕ b) = \frac{1}{2}(a + b). Then B is simple and purely infinite. Moreover, denoting by E = E_{(ℂ⊕ℂ)⊗O_{2}}^{B} the conditional expectation from B onto (ℂ ⊕ ℂ) ⊗ O_{2} arising from the free product construction, there exists a subalgebra M ⊂ B, so that:

(i) M ≅ M_{2}(ℂ) in such a way that the element (a ⊕ b) ⊗ 1 corresponds to the diagonal

matrix ( \begin{matrix}a & 0 \\ 0 & b \end{matrix}),

(ii) E(M) ⊂ (ℂ ⊕ ℂ) ⊗ 1.

Proof. Since O_{2} has trivial K–theory, it follows from Germain’s exact sequence for free products (see §2.3) that K_{0}(B) is zero. Once B is known to be simple and purely infinite, it will follow from Cuntz’s fundamental results [15], that there exists a partial isometry w ∈ B so that ww^{∗} = (1 ⊕ 0) ⊗ 1 and w^{∗}w = (0 ⊕ 1) ⊗ 1. Set M = C^{∗}(w). Then M ≅ M_{2}(ℂ), in such a way that w corresponds to the partial isometry ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), and therefore (i) is satisfied. To see that (ii) is satisfied, note that M is the linear span of w, w^{∗}, ww^{∗}, w^{∗}w; hence it is sufficient to check that E(w), E(w^{∗}w), E(ww^{∗}) and E(w^{∗}) = E(w)^{∗} all lie in (ℂ ⊕ ℂ) ⊗ 1. This is clearly true of E(w^{∗}w) and E(ww^{∗}), since ww^{∗} and w^{∗}w lie in (ℂ ⊕ ℂ) ⊗ 1. Furthermore, since (ww^{∗})E(w) = E(ww^{∗}w) = E(w)w^{∗}w), and the projections ww^{∗} and w^{∗}w are orthogonal, it follows that ww^{∗}E(w)ww^{∗} and w^{∗}wE(w)w^{∗}w are both zero. Since ww^{∗} and w^{∗}w lie in the center of (ℂ ⊕ ℂ) ⊗ O_{2} and ww^{∗} + w^{∗}w = 1, it follows that E(w) = 0, and hence E(w^{∗}) = 0, so that (ii) is satisfied as well.

Note that B ≅ (O_{2}⊕O_{2}, σ′)∗(C, ρ), where σ′ = \frac{1}{2}(σ+σ). We shall now apply Theorem 3.1 of [17] to show that B is simple and purely infinite; the following five observations show that the hypotheses of this theorem are satisfied.

1. σ′ and ρ are faithful states.

2. Let v = S_{2}S_{2}S_{1}^{∗} ⊕ 0 ∈ O_{2} ⊕ O_{2}; then v is a partial isometry belonging to the spectral subspace of the state σ′ ∗ ρ corresponding to 2, i.e. F (xv) = 2F (vx) for all x ∈ B.

3. Let q_{1} = v^{∗}v and q_{2} = vv^{∗}; then q_{1} ⊥ q_{2} and we have F (q_{1}) = \frac{1}{4}, F (q_{2}) = \frac{1}{8}.

4. Consider

B_{1} = C^{∗}(q_{1}, q_{2}, u) ⊆ B
B_{2} = C^{∗}(uq_{1}u^{∗}, q_{2}) ⊆ B_{1}.

By Lemma 5.1, uq_{1}u^{∗} and q_{2} are free with respect to F, so

B_{2} ≅ ( ℂ_{3/4} ⊕ ℂ_{1/4}^{uq_{1}u^{∗}}) ∗ ( ℂ_{7/8} ⊕ ℂ_{1/4}^{q_{2}})

and it is well known (cf. Proposition 2.7 of [16]) that

B_{2} ≅ ℂ ⊕ C([0, 1], M_{2}(ℂ)) ⊕ ℂ

with

uq_{1}u^{∗} ∼ 1 ⊕ ( \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ) ⊕ 0
q_{2} ∼ 0 ⊕ ( \begin{matrix}t & \sqrt{t(1−t)} \\ \sqrt{t(1−t)} & 1−t \end{matrix} ) ⊕ 0.

Therefore, q_{2} is equivalent in B_{1} to a subprojection of q_{1} and q_{2}B_{1}q_{2} contains the diffuse abelian subalgebra q_{2}B_{2}q_{2}.

5. The centralizer of σ′ contains a diffuse abelian subalgebra; hence by Proposition 3.2 of [16], B is simple and thus q_{1} + q_{2} is full in B.

The above facts allow us to apply Theorem 3.1 of [17], and we conclude that B is simple and purely infinite.

Lemma 5.5. Let W be a unital C^{∗}–algebra and let E_{W}^{X} : X → W and E_{W}^{Y} : Y → W be W –probability spaces, with automorphisms α_{X} and α_{Y}, respectively, which agree on W. Assume that W ⊂ Z ⊂ X is an α_{X}–invariant subalgebra, and let E_{W}^{Z} be the restriction of E_{W}^{X} to Z. Assume that there exists an E_{W}^{X} –preserving α_{X}–invariant conditional expectation E_{Z}^{X} : X → Z, and assume that the GNS representations associated to E_{W}^{X} , E_{W}^{Z} , E_{Z}^{X} and E_{W}^{Y} are faithful. Then there exists an isomorphism of the reduced free products

X ∗_{Z} (Z ∗_{W} Y) ≅ X ∗_{W} Y,

which intertwines the automorphisms α_{X} ∗ (α_{X}|_{Z} ∗ α_{Y}) and α_{X} ∗ α_{Y}.

Proof. Consider the free product conditional expectation E_{Z} : (X ∗_{Z} (Z ∗_{W} Y) → Z. The GNS representation associated to E_{Z} is faithful, by definition. Let E_{W} = E_{W}^{Z} ∘ E_{Z}. By Lemma 5.2, E_{W} also gives rise to a faithful GNS representation. By Lemma 2.6 in [36], X and Y are free with respect to E_{W}. It follows from the assumptions and the embedding result (see §2.2) that the C^{∗}–algebra generated by X and Y in the GNS representation of X ∗_{Z} (Z ∗_{W} Y) associated to E_{W} is isomorphic to X ∗_{W} Z. The desired isomorphism follows.

Lemma 5.6. Let E_{i} : A_{i} → D, i = 1, 2 be D–probability spaces, with automorphisms α_{i}. Assume that the GNS representations associated to E_{i} are faithful, and that α_{1}|_{D} = α_{2}|_{D}.

Let E : A_{1} ⊕ A_{2} → D = (d ⊕ d : d ∈ D) ⊂ A_{1} ⊕ A_{2} be given by

E(a_{1} ⊕ a_{2}) = \frac{1}{2}([E(a_{1}) + E(a_{2})] ⊕ [E(a_{1}) ⊕ E(a_{2})]).

Let σ : O_{2} → ℂ be as in Lemma 5.4. Let C be any C^{∗}–algebra with a state ρ̂, giving rise to a faithful GNS representation, and containing a unitary u, s.t. ρ = ρ̂|_{C^{∗}(u)} is faithful and ρ(u) = 0.

Then there exists an embedding

λ : (A_{1}, E_{1}) ∗_{D} (A_{2}, E_{2}) → B = ((A_{1} ⊕ A_{2}) ⊗ O_{2}), E ⊗ σ) ∗_{D} (C ⊗_{min} D, ρ̂ ⊗ id),

so that λ intertwines (α_{1} ∗ α_{2}) and ((α_{1} ⊕ α_{2}) ⊗ id) ∗ (id ⊗α_{1}|_{D}).

Proof. Using the embedding result (see §2.2), we can reduce to the case that C = C^{∗}(u) and ρ̂ = ρ, by replacing B with the algebra generated by (A_{1} ⊕A_{2})⊗O_{2} and C^{∗}(u)⊗D ⊂ C ⊗_{min} D.

We have the following sequence of covariant inclusions, justified below:

A_{1} ∗_{D} A_{2} ↪ M_{2} ⊗ (A_{1} ∗_{D} A_{2})
≅^{Lemma 5.3} M_{2} ⊗ D ∗_{D⊕D} (A_{1} ⊕ A_{2})
↪^{(a)} ([(ℂ^{2}) ⊗ O_{2}) ∗ C] ⊗ D) ∗_{(ℂ^{2})⊗O_{2}⊗D} [(A_{1} ⊕ A_{2}) ⊗ O_{2}]
≅ ((ℂ^{2} ⊗ O_{2} ⊗ D) ∗_{D} (C ⊗ D)) ∗_{(ℂ^{2})⊗O_{2}⊗D} [(A_{1} ⊕ A_{2}) ⊗ O_{2}]
≅^{(b)} C ⊗ D ∗_{D} [(A_{1} ⊕ A_{2}) ⊗ O_{2}].

Inclusion (a) is implied by Lemma 5.4, together with the embedding result (see §2.2). Isomorphism (b) is implied by Lemma 5.5.

Theorem 5.7. Let D be a finite dimensional C^{∗}–algebra and let α_{j}, be an automorphism of a D–probability space (A_{j}, E_{j} : A_{j} → D), (j ∈ J, J a finite or countably infinite set). Assume that the GNS representations associated to E_{j} (j ∈ J) are faithful. Assume that α_{j}|_{D} = α_{i}|_{D} for all i, j ∈ J. Let (A, E) = ∗_{D} ((A_{j}, E_{j}), j ∈ J) and let ∗_{j}α_{j} denote the free product automorphism of A. Then

ht(∗_{j}α_{j}) = sup_{j∈J}(ht(α_{j})). (5.1)

Proof. Because of the embedding result (see §2.2) and the behavior of entropy with respect to inductive limits, it suffices to prove the statement when J = {1, 2} is a set with two elements.

By monotonicity of ht, the inequality ≥ in (5.1) is clear. Let T be the Toeplitz algebra generated by the unilateral shift ℓ and denote by ρ̂ the vacuum expectation on T . Voiculescu [42] showed that ℓ^{∗} + ℓ is a semicircular element with its spectrum an interval. Hence by functional calculus C = C^{∗}(ℓ^{∗} + ℓ) contains a unitary u such that ρ̂(u^{k}) = 0, for all k ≠ 0. It is not difficult to see that C contains no nonzero compact operator, and thus ρ = ρ̂|_{C} is faithful. By Lemma 5.6 and Lemma 5.3, the amalgamated free product dynamical system (A_{1} ∗_{D} A_{2}, α_{1} ∗ α_{2}) can be covariantly embedded into (B, β), where

(B, F) = ((A_{1} ⊕ A_{2}) ⊗ O_{2}, Ẽ ⊗ σ) ∗_{D} (T ⊗ D, ρ̂ ⊗ id_{D})

and where

β = ((α_{1} ⊕ α_{2}) ⊗ id_{O_{2}}) ∗ (id_{T} ⊗(α_{1}|_{D})).

So by monotonicity again, ht(α_{1} ∗ α_{2}) ≤ ht(β). Using Theorem 4.1 we have

ht(β) = ht((α_{1} ⊕ α_{2}) ⊗ id_{O_{2}}) = ht(α_{1} ⊕ α_{2}) = max(ht(α_{1}), ht(α_{2})).

Remarks 5.8.

5.8.1. The only reason that the hypotheses of Theorem 5.7 require finite dimensionality of D is because the proof appeals to Theorem 4.1. If one could prove a version of Theorem 4.1 for more general types of D, then Theorem 5.7 would be valid for those more general algebras D, with the same proof.

5.8.2. Theorem 5.7 also holds for amalgamated free products of injective endomorphisms of non–commutative probability spaces. This can be seen by realizing an injective endomorphism as the restriction of an automorphism (see e.g. [20]), and then utilizing our result for automorphisms.

5.8.3. One can actually prove a partial version of Theorem 5.7 with amalgamation taking place over an AF algebra. Assume that E_{1} : A_{1} → D and E_{2} : A_{2} → D are D–probability spaces, with automorphisms α_{1}, α_{2}, which restrict to the same automorphism on D. Assume that D = \overline{∪_{i}D^{(i)}}, with D^{(i)} ⊂ D^{(i+1)} finite dimensional. Assume further that A_{j} =

\overline{∪_{i}A_{j}^{(i)}}, A_{j}^{(i)} ⊂ A_{j}^{(i+1)}, and that E_{j}(A_{j}^{(i)}) = D^{(i)} and D^{(i)} ⊂ A_{j}^{(i)}, j = 1, 2. Let α_{1}∗α_{2} denote the free product automorphism on A_{1}∗_{D}A_{2}. Then ht(α_{1}∗α_{2}) = max(ht(α_{1}), ht(α_{2})). This is due to the fact that under these assumptions, A_{1} ∗_{D} A_{2} is the direct limit of A_{1}^{(i)} ∗_{D^{(i)}} A_{2}^{(i)}.

6. Applications

6.1. Shifts on infinite free products and twisted free permutations. Let A be a unital C^{∗}–algebra, D a finite dimensional unital subalgebra and let E : A → D be a conditional expectation with a faithful GNS representation. Let I be a nonempty set and for every i ∈ I let (A_{i}, E_{i}), be a copy of (A, E). Let

(B, F) = (∗_{D})_{i∈I}(A_{i}, E_{i})

be the reduced amalgamated free product. Let σ : I → I be a bijection. The free permutation arising from σ is the automorphism β of B that permutes the copies of A inside B by sending A_{i} to A_{σ(i)}. (The free shift is the free permutation arising from the shift on I = ℤ.) The following theorem is a generalization of results of Størmer [39], Brown and Choda [8] and Dykema [18] and a partial generalization of other results of Størmer, [40]. It answers affirmatively in the case of amalgamation over a finite dimensional C^{∗} algebra Question 11 of [18]:

Theorem 6.1.1. If A is an exact unital C^{∗}–algebra and D is finite dimensional then for every free permutation β of B = (∗_{D})_{i∈I}A one has ht(β) = 0.

Proof. Consider the reduced group C^{∗}–algebra C_{r}^{∗}(𝔽_{|I|}) of the nonabelian free group on |I| generators, where |I| is the cardinality of I. Let (u_{i})_{i∈I} be the unitary generators of C_{r}^{∗}(𝔽_{|I|}) corresponding to the free generators of 𝔽_{|I|}. Let σ_{∗} ∈ Aut(C_{r}^{∗}(𝔽_{|I|})) be the automorphism such that σ_{∗}(u_{i}) = u_{σ(i)}. Then from [8] and [18], ht(σ_{∗}) = 0, Let

(B̃, F̃) = (C_{r}^{∗}(𝔽_{|I|}) ⊗ D, τ ⊗ id_{D}) ∗_{D} (A, E),

where τ is the canonical tracial state on C_{r}^{∗}(𝔽_{|I|}), and let

β̃ = (σ_{∗} ⊗ id_{D}) ∗ id_{A} ∈ Aut(B̃).

Then by Theorem 5.7, ht(β̃) = ht(σ_{∗}) = 0. Let B′ = C^{∗}(⋃_{i∈I} u_{i}Au_{i}^{∗}) ⊆ B̃. Then F̃(u_{i}au_{i}^{∗}) = E(a) for all a ∈ A and i ∈ I and, by Lemma 5.1, the family (u_{i}Au_{i}^{∗})_{i∈I} is free with respect to F̃. It is not difficult to see, though somewhat tedious to write down in detail, that the inclusion representation of B′ on L^{2}(B̃, F̃) is a multiple of the GNS representation, ρ, of B′ on L^{2}(B′, F̃|_{B′}). Indeed, one chooses vectors ξ_{i} ∈ L^{2}(B̃, F̃) such that ⋃_{i} B′ξ_{i} spans a dense subset of L^{2}(B̃, F̃) and so that the representation of B′ on B′ξ_{i} is equivalent to ρ, for all i. Hence there is an isomorphism π : B → B′ sending the copy of A_{i} in B to u_{i}Au_{i}^{∗}. We see that the automorphisms β and β̃|_{B′} are conjugate via π. Hence, by monotonicity of ht we have ht(β) ≤ ht(β̃) = 0.

Remark 6.1.2. Just as in 5.8.3, it is possible to weaken the hypothesis of the preceding Theorem by requiring that (A, D, E : A → D) is an inductive limit of (A^{(i)}, D^{(i)}, E_{i}), with D^{(i)} finite dimensional.

Definition 6.1.3. Let D be a unital C^{∗}–algebra, let I be a set and for every i ∈ I let (A_{i}, E_{i}) be a D–probability space such that the GNS representation associated to E_{i} is faithful. Let

(A, E) = (∗_{D})_{i∈I}(A_{i}, E_{i})

be the reduced amalgamated free product. Let σ : I → I be a bijection such that for every i ∈ I there is an isomorphism α_{i} : A_{i} → A_{σ(i)} such that α_{i}(D) = D and E_{σ(i)} ∘ α_{i} = E_{i}. Assume that α_{i}|_{D} is the same for all i. Then there is an automorphism α of A sending the copy of A_{i} in A onto the copy of A_{σ(i)} in A via α_{i}. An automorphism α arising in this way is called a twisted free permutation of A.

The next theorem generalizes Theorem 6.1.1 and also some results of [18].

Theorem 6.1.4. Suppose that D is finite dimensional and α ∈ Aut(A) is a twisted free permutation as described above. If the permutation σ of I has no cycles then ht(α) = 0. Otherwise, whenever c is a cycle of σ let ℓ(c) be the length of the cycle, let i ∈ I be one of the elements moved by the cycle and let β_{c} ∈ Aut(A_{i}) be the restriction of α^{ℓ(c)} to the copy of A_{i} in A. (Note that β_{c} depends on i only up to conjugation.) Then

ht(α) = sup_{c} \frac{ht(β_{c})}{ℓ(c)}, (6.1)

where the supremum is over all cycles c of σ.

Proof. If σ has no cycles then α is (conjugate to) a free permutation so ht(α) = 0 by Theorem 6.1.1. In general, by making a cycle decomposition of σ and using Theorem 5.7, we see that in order to prove (6.1) we may without loss of generality assume that σ itself is a cyclic permutation, σ = c, of a finite set I. However, then α^{ℓ(c)} = ∗_{i∈I}γ_{i}, where each γ_{i} ∈ Aut(A_{i}) is conjugate to β_{c}. Hence again applying Theorem 5.7 we have ht(α) = ht(α^{|I|})/|I| = ht(β_{c})/ℓ(c).

6.2. The CNT variational principle. In classical ergodic theory an important result connecting topological and measurable entropy is the variational principle.

Definition 6.2.1. Let (A, α) be a unital exact C^{∗}–dynamical system. We say that (A, α) satisfies a CNT–variational principle if

ht(α) = sup_{ϕ} h_{ϕ}(α),

where the supremum is taken over all α–invariant states on A and h_{ϕ}(α) denotes the CNT– entropy of α with respect to ϕ (cf. [12]).

By [18, Prop. 9] we always have the inequality

ht(α) ≥ sup_{ϕ} h_{ϕ}(α).

Not every C^{∗}–dynamical system satisfies a CNT–variational principle. In [29] an example of a highly non–asymptotically abelian system was given for which ht(α) ≥ (log 2)/2 while h_{ϕ}(α) = 0 for the unique α–invariant (tracial) state. But, since ht(·) agrees with classical topological entropy and h_{ϕ}(·) agrees with classical Kolmogorov–Sinai entropy when A is abelian, the classical variational principle says that if A is unital and abelian then (A, α) always satisfies a CNT–variational principle. The list of examples of noncommutative dynamical systems which satisfy a CNT–variational principle is also rapidly growing (cf. [44, 4.7], [10, 4.6, 4.7], [7, 3.6, 3.7], [5], [34], [30]). Moreover, the class of C^{∗}–dynamical systems which satisfy a CNT–variational principle is closed under taking (minimal) tensor products (cf. [41, Lemma 3.4]) and crossed products by ℤ (cf. [7, Theorem 3.5]). Unfortunately it is not closed under taking quotients or subalgebras (even those with a conditional expectation onto them — simply take a direct sum of something abelian with large entropy and the example from [29]) and it not yet known what happens in extensions. However we now show that it is also closed under taking reduced free products. The next theorem also gives lots of examples of non–asymptotically abelian dynamical systems for which the CNT–variational principle holds.

Theorem 6.2.2. Let E_{j} : A_{j} → D, j = 1, 2 be non–commutative probability spaces with automorphisms α_{1} and α_{2}. Assume that D is finite dimensional and that α_{1}|_{D} = α_{2}|_{D}, and that E_{j} give rise to faithful GNS representations. If both (A_{1}, α_{1}) and (A_{2}, α_{2}) satisfy the CNT variational principle, then so does (A_{1} ∗_{D} A_{2}, α_{1} ∗ α_{2}).

Proof. Assume without loss of generality that ht(α_{1}) ≥ ht(β_{1}). Given ϵ > 0 we can find an α_{1}–invariant state γ ∈ S(A_{1}) such that ht(α_{1}) ≤ h_{γ}(α_{1}) + ϵ.

Let E : A_{1} ∗ A_{2} → A_{1} be the canonical conditional expectation and define a state γ̃ = γ ∘ E ∈ S(A_{1} ∗_{D} A_{2}). Then one checks that γ̃ ∘ (α_{1} ∗ α_{2}) = γ̃ and γ̃ ∘ E = γ̃. But under these conditions, CNT–entropy is also monotone (cf. [12, III.6]) and so ht(α_{1}∗α_{2}) = ht(α_{1}) ≤ h_{γ}(α_{1}) + ϵ ≤ hγ̃(α_{1} ∗ α_{2}) + ϵ. Since ϵ was arbitrary, we are done.

A particularly interesting class of dynamical systems for which one would like to have a CNT–variational principle is those arising from automorphisms of discrete groups. For any discrete group G let C_{r}^{∗}(G) be the reduced group C^{∗}–algebra and τ_{G} the canonical trace on C_{r}^{∗}(G). G is called exact if C_{r}^{∗}(G) is exact. If γ : G → G is a group automorphism then there is an induced automorphism γ̂ ∈ Aut(C_{r}^{∗}(G)) such that τ_{G} ∘ γ̂ = τ_{G}. If G is abelian then a classical theorem of Berg [2] implies that ht(γ̂) = h_{τ_{G}}(γ̂).

Theorem 6.2.3. Let H be a finite group and let G_{1} and G_{2} be discrete exact groups having a H as a common subgroup. Suppose that γ_{i} ∈ Aut(G_{i}), (i = 1, 2) are automorphisms preserving H such that γ_{1}|_{H} = γ_{2}|_{H}. Let γ_{1} ∗ γ_{2} denote the resulting automorphism of the free product G_{1} ∗_{H} G_{2} with amalgamation over H. If ht(γ̂_{i}) = h_{τ_{G_{i}}}(γ̂_{i}), (i = 1, 2), then ht((γ_{1} ∗ γ_{2}) ) = h_{τ_{G_{1}∗_{H} G_{2}}}((γ_{1} ∗ γ_{2}) ).

Corollary 6.2.4. If G_{1}, G_{2} are discrete exact groups with automorphisms γ_{i} ∈ Aut(G_{i}), (i = 1, 2), and if ht(γ̂_{i}) = h_{τ_{G_{i}}}(γ̂_{i}) ,(i = 1, 2), then ht(γ̂_{1} ∗ γ̂_{2}) = h_{τ_{G_{1}∗G_{2}}}(γ̂_{1} ∗ γ̂_{2}).

It would be interesting to know whether or not the above theorem or its corollary can be extended to the dual entropy defined in [9] as well.

6.3. Entropy–preserving embeddings. Kirchberg first proved that every separable exact C^{∗}–algebra is isomorphic to a subalgebra of the Cuntz algebra on two generators (cf. [27]). In [7, Remark 2.3] it was asked whether or not one can always find a unital embedding ρ : A⋊_{α}ℤ ↪ O_{2} such that ht_{O_{2}}(Adρ(u)) = ht(α), where u ∈ A⋊_{α}ℤ is the implementing unitary. We now solve this problem affirmatively in the case that A is nuclear and there exists an α–invariant state ϕ ∈ S(A) with faithful GNS representation. We also show that if (A, α) is any nuclear C^{∗}–dynamical system then there always exists an entropy preserving covariant embedding into the Cuntz algebra on infinitely many generators. Since many C^{∗}–algebras are stable under tensor products with O_{∞} we thus get entropy preserving embeddings into a very large class of C^{∗}–algebras. It follows that the topological entropy invariant of all such algebras is [0, ∞].

We begin with a simple proposition.

Proposition 6.3.1. Assume A is a unital exact C^{∗}–algebra, α ∈ Aut(A) and there exists an α–invariant state ϕ ∈ S(A) with faithful GNS representation. Let B be unital and exact, let ψ ∈ S(B) have faithful GNS representation, let E : A ⋊_{α} ℤ → A be the canonical conditional expectation and let u ∈ A ⋊_{α} ℤ be the implementing unitary. Regarding u as a unitary in (A ⋊_{α} ℤ, ϕ ∘ E) ∗ (B, ψ) we have ht(Adu) = ht(α).

Proof. Since E : A ⋊_{α} ℤ → A is faithful, by Lemma 5.2 the GNS representation of ϕ ∘ E is faithful.

Consider the C^{∗}–algebra (A, ϕ) ∗ ( ∗_{ℤ} (B, ψ)) ⋊_{α∗S} ℤ, where S : ∗_{ℤ}(B, ψ) → ∗_{ℤ}(B, ψ) is the free shift and let v ∈ (A, ϕ) ∗ ( ∗_{ℤ} (B, ψ)) ⋊_{α∗S} ℤ be the implementing unitary. It is fairly easy to see (cf. the proof of [11, Claims 2 and 4]) that there exists a ∗–isomorphism

ρ : (A ⋊_{α} ℤ, ϕ ∘ E) ∗ (B, ψ) → (A, ϕ) ∗ (( ∗_{ℤ} (B, ψ)) ⋊_{α∗S} ℤ),

such that ρ(u) = v. Hence ht(Adu) = ht(Adv). But [7, Theorem 3.5], Theorem 6.1.1 above and our main result imply that ht(Adv) = ht(α ∗ S) = ht(α).

Theorem 6.3.2. Let A be a unital separable nuclear C^{∗}–algebra and α ∈ Aut(A) be an automorphism such that there exists an α–invariant state ϕ ∈ S(A) with faithful GNS representation. If O_{2} denotes the Cuntz algebra on two generators then there exists a unital embedding ρ : A ⋊_{α} ℤ ↪ O_{2} such that ht_{O_{2}}(Ad ρ(u)) = ht(α), where u ∈ A ⋊_{α} ℤ is the implementing unitary.

Proof. Replacing (A, α) with (A ⊗ C(𝕋), α ⊗ id_{C(𝕋)}), if necessary, we may assume that there exists a Haar unitary in the centralizer of ϕ. In this setting, [16, Prop. 3.2] ensures that (A ⋊_{α} ℤ, ϕ ∘ E) ∗ (T , v) is a simple C^{∗}–algebra, where T is the Toeplitz algebra and v is the vacuum state. Moreover, this free product is nuclear since it is isomorphic to a Cuntz–Pimsner algebra over A ⋊_{α} ℤ. Thus Kirchberg’s absorption theorem for O_{2} (i.e. B ⊗ O_{2} ≅ O_{2}, for any simple, separable, unital, nuclear C^{∗}–algebra B, cf. [27]) together with the previous proposition implies the result.

We now turn to covariant entropy preserving embeddings into O_{∞}.

Theorem 6.3.3. Let A be any separable nuclear C^{∗}–algebra and α ∈ Aut(A) be an automorphism. If O_{∞} denotes the Cuntz algebra on infinitely many generators then there exists an automorphism β ∈ Aut(O_{∞}) and a nonunital ∗–monomorphism π : A → O_{∞} such that β ∘ π = π ∘ α and ht(β) = ht(α).

Proof. By adding a unit to A and replacing A by its crossed product by α, we may assume that A is unital and α is inner. We may also assume that there exists an α–invariant state ϕ ∈ S(A) with faithful GNS representation. Indeed, it follows from Theorem 3.5 that the covariant embedding (A, α) ↪ (A ⊕ A + K(H ⊕ H), Ad(u ⊕ 1_{A})), where we regard A ⊂ B(H) and u ∈ A is a unitary which implements α, is entropy preserving. Now any vector state arising from the second copy of H will be Ad(u ⊕ 1_{A})–invariant and have faithful GNS representation since it is a cyclic vector.

So, we assume that A is unital, α is inner and ϕ ∈ S(A) is α–invariant with faithful GNS representation. Consider the free product

B = ((A ⊗ C[0, 1] ⊗ O_{2} ⊕ C[0, 1], η)) ∗ (T , v).

Here η is the average of Lebesgue measure on the second copy of C[0, 1] and the tensor product of ϕ, Lebesgue measure and an arbitrary faithful state on O_{2}, and v is the vacuum state on the Toeplitz algebra T . Since there is a Haar unitary in the centralizer of η (coming from the copies of C[0, 1]), [16, Prop. 3.2] implies that B is simple. Moreover, B is nuclear being isomorphic to a Cuntz–Pimsner algebra over A ⊗ C[0, 1] ⊗ O_{2} ⊕ C[0, 1] (cf. [20], [24]). Since O_{2} is KK–equivalent to zero, A⊗C[0, 1]⊗O_{2}⊕C[0, 1] satisfies the Universal Coefficient Theorem. Since T is type I it also satisfies the UCT and hence, by a result of Germain [23] so does B. Moreover, it follows from Germain’s six term exact sequence for K–theory (see §2.3) that B has the K–theory of O_{∞}. Hence B ⊗ O_{∞} is a simple, unital, purely infinite, nuclear C^{∗}–algebra which satisfies the UCT and has the K–theory of O_{∞}. So, by the classification results of Kirchberg–Phillips ([26], [32]), B ⊗ O_{∞} ≅ O_{∞}. By our main result we have that the automorphism γ = (α ⊗ id_{C[0,1]} ⊗ id_{O_{2}} ⊕ id_{C[0,1]}) ∗ id_{T} ∈ Aut(B) has the same entropy as α. Hence defining β = γ ⊗ id_{O_{∞}} ∈ Aut(B ⊗ O_{∞}) = Aut(O_{∞}) we get the desired entropy preserving covariant embedding.

Question 6.3.4. Let u ∈ A be a unitary and regard Ad u as an automorphism of (A, ϕ) ∗ (B, ψ). As in Proposition 6.3.1, is it true that ht_{A∗B}(Ad u) = ht(Ad u|_{A})? Clearly, an affirmative answer to this question would imply that the automorphism in Theorem 6.3.3 can be taken to be inner.

6.4. Possible values of entropy of all automorphisms of a C^{∗}–algebra.

Definition 6.4.1. If A is an exact C^{∗}–algebra then put

T E(A) = {t ∈ [0, ∞] : there existsα ∈ Aut(A) such that ht(α) = t}.

For A unital, we can also consider the set

T E_{Inn}(A) = {t ∈ [0, ∞] : there exists a unitaryu ∈ A such that ht(Adu) = t}.

The results of §6.3 imply, in particular, that T E(O_{∞}) = [0, ∞] and T E_{Inn}(O_{2}) = [0, ∞], since there are automorphisms of nuclear (in fact, abelian) algebras with any prescribed entropy. (See [6] for some nice examples.) Clearly, if B contains a projection, then T E(B ⊗ A) ⊃ T E(A). If A and B are both unital, we have T E_{Inn}(A ⊗ B) ⊃ T E_{Inn}(A) ∪ T E_{Inn}(B).

If B is any nuclear simple separable purely infinite C^{∗}–algebra then Kirchberg has shown that B ⊗ O_{∞} ≅ B (see [27]). Recently Kirchberg and Rørdam introduced a class of nonsimple purely infinite C^{∗}–algebras (cf. [28]). Though it is not yet known whether all of their nuclear purely infinite C^{∗}–algebras will absorb O_{∞}, many examples are known.

Theorem 6.4.2. Let B be any exact C^{∗}–algebra which contains a projection. Then T E(B ⊗ O_{∞}) = [0, ∞]. If B is unital then T E_{Inn}(B ⊗ O_{2}) = [0, ∞].

In particular, if B is nuclear, simple, separable and purely infinite then T E(B) = [0, ∞].

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Department of Mathematics, University of California, Berkeley, CA 94720

  E-mail address: nbrown@math.berkeley.edu

Department of Mathematics, Texas A&M University, College Station, TX 77843

  E-mail address: Ken.Dykema@math.tamu.edu

Department of Mathematics, University of California, Los Angeles, CA 90095

  E-mail address: shlyakht@math.ucla.edu