Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

Consider a uniquely ergodic C*-dynamical system based on a unital *-endomorphism Φ of a C*-algebra. We prove the uniform convergence of Cesaro averages 1n∑k=0n−1λ−nΦ(a) for all values λ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.


Introduction
Motivated by the question of justifying the thermodynamical laws with the microscopic principles of statistical mechanics (i.e., the so-called ergodic hypothesis), the investigation of the ergodic properties of classical (i.e., commutative) dynamical systems has a long history.
Indeed, given a classical dynamical system (X, T, µ), where X is a compact space, T : X → X a continuous map, and finally, µ an invariant probability measure under the natural action of T, the classical ergodic theory primarily deals with the long time behavior of the Cesaro means (ergodic averages) M n ( f ) := 1 n n−1 ∑ k=0 f • T k , f ∈ C(X) , n ∈ N , of continuous functions, or more generally of any measurable function f w.r.t. the σ-algebra generated by the µ-measurable sets. Among the most famous classical ergodic theorems, we mention the Birkhoff individual ergodic theorem concerning the study of the point-wise limit lim n→+∞ M n ( f )(x) and the von Neumann mean ergodic theorem concerning the limit L 2 −lim n→+∞ M n ( f ), whenever f is square-summable.
The quantity of results obtained in the commutative setting is too huge to summarize an exhaustive description. However, a standard reference, dealing mainly with the classical case, is [1]. We also mention several unconventional ergodic theorems (e.g., [2]), which play a fundamental role in number theory.
At the same way, also the investigation of the uniform convergence of ergodic averages (i.e., involving directly continuous functions in the commutative C * -algebra C(X)) is of great interest.
Among such kind of results, we mention the following one relative to the so-called uniquely ergodic dynamical systems. The classical dynamical system (X, T) is said to be uniquely ergodic if there exists a unique probability Radon measure µ, which is invariant under the action of the transformation T. It was proven in [4] that (X, T) is uniquely ergodic if and only if, for the Cesaro average of any f ∈ C(X), uniformly. In [5], the last result was generalized to averages of the form for certain λ in the unit circle T.
With the impetuous growth of quantum physics, it was natural to address the systematic investigation of ergodic properties of quantum (i.e., noncommutative) dynamical systems. On the other hand, the situation in the quantum setting appears rather more complicated than the classical situation. Typically, one must provide all statements in terms of the dual concept of "functions" instead of "points". Therefore, algebras of functions are replaced by general C * or W * -algebras A, and the action on functions Φ T ( f ) := f • T of the transformation T is replaced by that of a positive linear map Φ : A → A acting directly on elements of A. Concerning some general ergodic properties of noncommutative dynamical systems, the reader is referred to [6] and the literature cited therein.
The systematic study of some natural generalizations of ergodic properties to the quantum case has been carried out in the seminal paper [7]. The reader is also referred to [8][9][10][11] for some quantum versions of unconventional (called also "entangled") ergodic theorems and to [12][13][14] for the investigation of the strong ergodic properties of dynamical systems arising from free probability and generalizing the unique ergodicity. Some natural applications of ergodic results to quantum probability are also carried out; see [15] and the references cited therein.
The goal of the present note is to provide the quantum generalization of the interesting result proven in [5] involving the uniform convergence of Cesaro averages relative to uniquely ergodic quantum dynamical systems "continuous" eigenfunctions. This result can be considered a combination of the Wiener-Wintner theorem (cf. [16]) and the uniformly convergent ergodic theorem of Krylov and Bogolioubov (cf. [4]).
More precisely, let (A, Φ, ϕ) be a uniquely ergodic C * -dynamical system based on a unital C * -algebra and a unital * -homomorphism Φ : A → A with ϕ ∈ S(A) as the unique invariant state. Consider the covariant Gelfand-Naimark-Segal representation H ϕ , π ϕ , V ϕ,Φ , ξ ϕ associated with the state ϕ, together with the peripheral pure-point spectra (see below for the definition) σ ph pp (Φ) and σ ph pp (V ϕ,Φ ) of Φ and the isometry V ϕ,Φ ∈ B(H ϕ ), respectively. We see that σ ph pp (Φ) ⊂ σ ph pp (V ϕ,Φ ), but in general, they are different. Put for a ∈ A and λ ∈ T, We show that, in the norm topology of A (compare with Proposition 3.2 in [7]), where u λ ∈ A is a unitary eigenvector (i.e., a "continuous eigenfunction" in the language of [5]) corresponding to λ, which is uniquely determined up to a phase-factor; , λ does not admit any nontrivial "measurable eigenfunction" in the language of [5]), then M a,λ (n) → 0.
We end the paper with some example based on the tensor product, which is however nontrivial, of an Anzai skew product (cf. [17]) and a uniquely mixing noncommutative dynamical system, for which the sequence M a,λ (n) n∈N does not converge for some a ∈ A and λ ∈ σ ph pp (V ϕ,Φ )\σ ph pp (Φ).
For ξ ∈ H ϕ and n ∈ Z, consider the sequence Proposition 1. For each ξ ∈ H ϕ , the sequence µ ξ (n) n∈Z is positive definite, and therefore, it is the Fourier transform of a positive bounded Radon measure µ ξ on the unit circle T.
The C * -dynamical system (A, Φ) made of a unital C * -algebra A and an identity-preserving completely positive map Φ : A → A is said to be uniquely ergodic if there exists only one invariant state ϕ for the dynamics induced by Φ. For a uniquely ergodic C * -dynamical system, we simply write (A, Φ, ϕ) by pointing out that ϕ ∈ S(A) is the unique invariant state.
From now on, we specialize the situation to the case when Φ is a unital * -homomorphism of the unital C * -algebra A.
For the sake of completeness, we collect some standard results, which are probably known to the experts. Proof. Since (A, Φ, ϕ) is uniquely ergodic, we have for the ergodic average, uniformly, where ϕ is the unique invariant state. Suppose a ∈ A Φ . We get a = 1 n ∑ n−1 k=0 Φ k (a) −→ ϕ(a)1 I, and thus, a is a multiple of the identity.
Fix now λ ∈ σ ph pp (Φ) and a, b ∈ A λ \{0}. Then, a * b = α1 I and ba * = β1 I for some numbers α, β. Suppose α = 0. Since aa * is a non-null multiple, say c, of the identity, we have aa * b = 0, which means b = 0, a contradiction. At the same way, we verify ba * = 0. Now, α −1 a * and β −1 a * are left and right inverses of b. This means that b is invertible and b −1 = α −1 a * . At the same way, a is invertible, as well. Moreover, ab −1 = α −1 aa * = α −1 cI. This means a = α −1 cb, that is a is a multiple of b. Since aa * = cI, we argue that A λ = Cu λ for the unitary u λ = c −1/2 a.
Let now λ j ∈ σ ph pp (Φ) with u λ j unitaries in A λ j , j = 1, 2. First, u * λ j is a unitary eigenvector corresponding to λ −1 j because Φ is a real map. Second, u λ 1 u λ 2 is a unitary eigenvector corresponding to λ 1 λ 2 because Φ is multiplicative.
Proof. Fix λ ∈ σ ph pp (Φ), together with a unitary eigenvector u λ ∈ A λ , which exists by the previous proposition. For η λ := π ϕ (u λ )ξ ϕ ∈ H ϕ , first, we get V ϕ,Φ η λ = λη λ , and second, η λ 2 = ϕ(u * λ u λ ) = 1. The key-point of our analysis is the following result, which is nothing but the noncommutative version of Lemma 2.1 in [5]. Lemma 1. Consider the uniquely ergodic C * -dynamical system (A, Φ, ϕ), together with a sequence of states {ω n } n∈N ⊂ S(A). Then, for each a ∈ A and λ = e −ıθ , Proof. With λ = e −ıθ , consider the C * -tensor product C(T) ⊗ A ≡ C(T; A) together with the * -homomorphism Φ : C(T; A) → C(T; A) given by ) be the sequence of states given by Notice that for the function f (s) := ae ıs ∈ C (T; A), Let {n j } j∈N ⊂ N be a subsequence such that lim sup and consider any * -weak limit point ω of the sequence { ω n j } j∈N , which exists by the Banach Alaoglu theorem, see, e.g., [19], Theorem 4.21. By passing to a subsequence if necessary, we get Let ω ∈ S(A) be the marginal of ω defined on constant functions f a (s) := a by By construction, ω is invariant under Φ. Therefore, ω is invariant under Φ, as well, which means ξ ω be the covariant GNS representation associated with ω. By computing as in Lemma 2.1 of [5], we then conclude for the spectral measures associated with V ω, Φ and V ϕ,Φ , Therefore, with P const ∈ B(H ω ), the orthogonal projection onto the one-dimensional subspace C1 I C(T;A) = C ⊗ 1 I A ,

The Main Result
The present section is devoted to the following ergodic result we want to prove. Denote with χ S the characteristic function of the subset S ⊂ X by uniformly for n → +∞, where u λ ∈ A λ is any unitary eigenvalue corresponding to λ ∈ σ ph pp (Φ).
Consider the free group F Z on infinitely many generators {g j } j∈Z , together with the one-step shift g j → g j+1 acting on the generators. Such a shift induces an action of the group of the integers Z on the reduced group C * -algebra C * red (F Z ) generated by all powers of the corresponding * -automorphism γ(λ(g j )) := λ(g j+1 ), λ(g j ) j∈Z ⊂ C * red (F Z ) being the unitary generators of the reduced group C * -algebra. Here, we have denoted by "λ" the left regular representation of the discrete group F Z on 2 (F Z ). The left regular representation also realises, up to unitary equivalence, the GNS representation of the reduced group C * -algebra associated to the canonical trace.
It was shown in Corollary 3.3 of [12] that the C * -dynamical system C * red (F Z ), γ is uniquely mixing, and thus uniquely ergodic with the canonical trace τ as the unique invariant state.
Denote by H τ , π τ , V τ,γ , ξ τ the GNS covariant representation associated with C * red (F Z ), γ, τ . In particular, we have σ ph pp (V τ,γ ) = {1} (Here, we have denoted by "λ" the left regular representation of the discrete group F Z on 2 (F Z ). The left regular representation also realizes, up to unitary equivalence, the GNS representation of the reduced group C * -algebra associated with the canonical trace.).

Conclusions
The possibility to address various generalizations of ergodic results to the quantum case is usually an improbate task. Concerning the present argument, it was still possible to extend the classical result mutatis-mutandis to the quantum situation. However, the following (stimulating in the opinion of the author) problems remain open: • investigate under which conditions the average (1) still converge, even for λ ∈ σ ph pp (V ϕ,Φ )\σ ph pp (Φ); • extend our main result (i.e., Theorem 1) to general positive maps Φ : A → A acting on the C * -algebra A; • provide more complex (i.e., which do not merely come from a tensor product construction) examples for which the average (1) does not converge for some λ ∈ σ ph pp (V ϕ,Φ )\σ ph pp (Φ).

Funding:
The author acknowledges the financial support relative to the project "sustainability: OAAMP-Algebre di operatori e applicazioni a strutture non commutative in matematica e fisica" CUPE81I18000070005, and of the Italian INDAM-GNAMPA. The present project is also part of "MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006".

Conflicts of Interest:
The author declares no conflict of interest.