All articles published by MDPI are made immediately available worldwide under an open access license. No special
permission is required to reuse all or part of the article published by MDPI, including figures and tables. For
articles published under an open access Creative Common CC BY license, any part of the article may be reused without
permission provided that the original article is clearly cited. For more information, please refer to
Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature
Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for
future research directions and describes possible research applications.
Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive
positive feedback from the reviewers.
Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world.
Editors select a small number of articles recently published in the journal that they believe will be particularly
interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the
most exciting work published in the various research areas of the journal.
Consider a uniquely ergodic -dynamical system based on a unital *-endomorphism of a -algebra. We prove the uniform convergence of Cesaro averages for all values 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.
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 , where X is a compact space, a continuous map, and finally, 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)
of continuous functions, or more generally of any measurable function f w.r.t. the -algebra generated by the -measurable sets.
Among the most famous classical ergodic theorems, we mention the Birkhoff individual ergodic theorem concerning the study of the point-wise limit and the von Neumann mean ergodic theorem concerning the limit , 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 . We also mention several unconventional ergodic theorems (e.g., ), which play a fundamental role in number theory.
Most of the known ergodic results concern the so-called -setting, which roughly speaking involve the functions in the commutative -algebra ; see, e.g., Theorem III.1.2 in .
At the same way, also the investigation of the uniform convergence of ergodic averages (i.e., involving directly continuous functions in the commutative -algebra ) 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 is said to be uniquely ergodic if there exists a unique probability Radon measure , which is invariant under the action of the transformation T. It was proven in  that is uniquely ergodic if and only if, for the Cesaro average of any ,
uniformly. In , the last result was generalized to averages of the form
for certain in the unit circle .
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 or -algebras , and the action on functions of the transformation T is replaced by that of a positive linear map acting directly on elements of . Concerning some general ergodic properties of noncommutative dynamical systems, the reader is referred to  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 . 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  and the references cited therein.
The goal of the present note is to provide the quantum generalization of the interesting result proven in  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. ) and the uniformly convergent ergodic theorem of Krylov and Bogolioubov (cf. ).
More precisely, let be a uniquely ergodic -dynamical system based on a unital -algebra and a unital *-homomorphism with as the unique invariant state. Consider the covariant Gelfand–Naimark–Segal representation associated with the state , together with the peripheral pure-point spectra (see below for the definition) and of and the isometry , respectively. We see that , but in general, they are different. Put for and ,
We show that, in the norm topology of (compare with Proposition 3.2 in ),
if , then , where is a unitary eigenvector (i.e., a “continuous eigenfunction” in the language of ) corresponding to , which is uniquely determined up to a phase-factor;
if (i.e., does not admit any nontrivial “measurable eigenfunction” in the language of ), then .
We end the paper with some example based on the tensor product, which is however nontrivial, of an Anzai skew product (cf. ) and a uniquely mixing noncommutative dynamical system, for which the sequence does not converge for some and .
With , we denote the unit circle of the complex plane. It is homeomorphic to the interval by , after identifying the end-points zero and .
A (discrete) -dynamical system is a triplet consisting of a -algebra, a positive map , and a state such that . Consider the Gelfand–Naimark–Segal (GNS for short) representation ; see, e.g., . If in addition
then there exists a unique linear contraction such that and
The quadruple is called the covariant GNS representation associated with the triplet .
If is multiplicative, hence a *-homomorphism, then is an isometry with final range , the orthogonal projection onto the subspace ; see, e.g., Lemma 2.1 of .
For the -dynamical system , the case when is a unital -algebra with unity , and is multiplicative and identity-preserving, i.e., a unital *-homomorphism, is of primary importance. Indeed, denote by the fixed-point subalgebra, and
the set of the peripheral eigenvalues of (i.e., the peripheral pure-point spectrum), with the relative eigenspaces. Obviously, .
For and , consider the sequence
For each , the sequence is positive definite, and therefore, it is the Fourier transform of a positive bounded Radon measure on the unit circle .
Since is multiplicative, is an isometry. Consider its Sz-Nagy dilation (cf. )
acting on the direct sum , together with its spectral resolution (e.g., )
and finally the vector
and therefore, is the Fourier transform of the positive bounded Radon measure on the unit circle. □
Consider the pure-point peripheral spectrum
of . Denote with the orthogonal projection onto the eigenspace generated by the eigenvectors associated with , with the convention if .
With an abuse of language, is the spectral measure of relative to . Therefore, if , then .
The -dynamical system made of a unital -algebra and an identity-preserving completely positive map is said to be uniquely ergodic if there exists only one invariant state for the dynamics induced by . For a uniquely ergodic -dynamical system, we simply write by pointing out that is the unique invariant state.
From now on, we specialize the situation to the case when is a unital *-homomorphism of the unital -algebra .
For the sake of completeness, we collect some standard results, which are probably known to the experts.
Let the -dynamical system be uniquely ergodic. Then, is a subgroup of , and all corresponding eigenspaces , , have dimension one and are generated by a single unitary .
Since is uniquely ergodic, we have for the ergodic average,
uniformly, where is the unique invariant state. Suppose . We get , and thus, a is a multiple of the identity.
Fix now and . Then, and for some numbers . Suppose . Since is a non-null multiple, say c, of the identity, we have , which means , a contradiction. At the same way, we verify . Now, and are left and right inverses of b. This means that b is invertible and . At the same way, a is invertible, as well. Moreover, . This means , that is a is a multiple of b. Since , we argue that for the unitary .
Let now with unitaries in , . First, is a unitary eigenvector corresponding to because is a real map. Second, is a unitary eigenvector corresponding to because is multiplicative. □
Let the -dynamical system be uniquely ergodic. Then, .
Fix , together with a unitary eigenvector , which exists by the previous proposition. For , first, we get , and second, . □
The key-point of our analysis is the following result, which is nothing but the noncommutative version of Lemma 2.1 in .
Consider the uniquely ergodic -dynamical system , together with a sequence of states . Then, for each and ,
With , consider the -tensor product together with the *-homomorphism given by
For , let be the sequence of states given by
Notice that for the function ,
Let be a subsequence such that
and consider any *-weak limit point of the sequence , which exists by the Banach Alaoglu theorem, see, e.g., , Theorem 4.21. By passing to a subsequence if necessary, we get
Let be the marginal of defined on constant functions by
By construction, is invariant under . Therefore, is invariant under , as well, which means because is uniquely ergodic.
Let be the covariant GNS representation associated with . By computing as in Lemma 2.1 of , we then conclude for the spectral measures associated with and ,
Therefore, with , the orthogonal projection onto the one-dimensional subspace ,
3. The Main Result
The present section is devoted to the following ergodic result we want to prove. Denote with the characteristic function of the subset by
Let be a uniquely ergodic -dynamical system. Fix . Then, for each ,
uniformly for , where is any unitary eigenvalue corresponding to .
First consider the case , and take a unitary eigenvector . Since is multiplicative, we have
Let now , and suppose uniformly. Then, there would exist a sequence of states states such that for , . By Lemma 1,
which contradicts . □
We end by constructing simple noncommutative examples for which by tensoring a uniquely mixing (see ) noncommutative -dynamical system based on the *-automorphism , with some Anzai skew product (cf. ) as those described in Section 3 of .
Consider the free group on infinitely many generators , together with the one-step shift acting on the generators. Such a shift induces an action of the group of the integers on the reduced group -algebra generated by all powers of the corresponding *-automorphism , being the unitary generators of the reduced group -algebra. Here, we have denoted by "" the left regular representation of the discrete group on . The left regular representation also realises, up to unitary equivalence, the GNS representation of the reduced group -algebra associated to the canonical trace.
It was shown in Corollary 3.3 of  that the -dynamical system is uniquely mixing, and thus uniquely ergodic with the canonical trace as the unique invariant state.
Denote by the GNS covariant representation associated with . In particular, we have (Here, we have denoted by “” the left regular representation of the discrete group on . The left regular representation also realizes, up to unitary equivalence, the GNS representation of the reduced group -algebra associated with the canonical trace.).
Let be the tensor product -dynamical system, where
Here, , and is the Anzai skew product corresponding to the rotation of the angle such that is irrational, and to the continuous function .
If the Anzai skew product T is uniquely ergodic, then the above -dynamical system is uniquely ergodic, as well.
In addition, there exist Anzai skew products T such that , and the limit
fails to exist in the weak topology, for some and .
Since is uniquely mixing and the Anzai skew product is supposed to be uniquely ergodic, by Theorem 3.7 of , we argue that is uniquely ergodic.
Notice that . Therefore, by Lemma 4.17 of , each eigenvector corresponding to the eigenvalue is of the form for some unitary function (i.e., a measurable eigenvector in the language of ), which is an eigenvector of the Anzai skew-product (i.e., ) corresponding to the same value of . If moreover, (i.e., a continuous eigenvector in the language of ), then , being the dual action of T on functions: . Summarizing, we have
In order to check the latter assertion, it is enough to consider an Anzai skew product and a continuous function as in Proposition 3.1 of , such that
fails to exist for the point and . Therefore, for the element and state , we get
which fails to exist. □
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 ;
extend our main result (i.e., Theorem 1) to general positive maps acting on the -algebra ;
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 .
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.
Krengel, U. Ergodic Theorems; De Gruyter Studies in Mathematics; Walter de Gruyter: Berlin, Germany, 1985; Volume 6. [Google Scholar]
Furstenberg, H. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. J. Anal. Math.1977, 31, 204–256. [Google Scholar] [CrossRef]
Takesaki, M. Theory of Operator Algebras I; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1979. [Google Scholar]
Krylov, N.; Bogoliubov, N. La théorie générale de la mesure dans son application à l’étude des systémes dynamiques de la mécanique non-linéaire. Ann. Math.1937, 38, 65–113. [Google Scholar] [CrossRef]
Robinson, E.A., Jr. On uniform convergence in the Wiener–Wintner theorem. J. Lond. Math. Soc.1994, 49, 493–501. [Google Scholar] [CrossRef]
Bratteli, O.; Robinson, D.W. Operator Algebras and Quantum Statistical Mechanics I, II; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1981. [Google Scholar]
Niculescu, C.P.; Ströh, A.; Zsidó, L. Noncommutative estension of classical and multiple recurrence theorems. J. Oper. Theory2003, 50, 3–52. [Google Scholar]
Fidaleo, F. On the entangled ergodic theorem. Infin. Dimens. Anal. Quantum Probab. Relat. Top.2007, 10, 67–77. [Google Scholar] [CrossRef]
Fidaleo, F. An ergodic theorem for quantum diagonal measures. Infin. Dimens. Anal. Quantum Probab. Relat. Top.2009, 12, 307–320. [Google Scholar] [CrossRef]
Fidaleo, F. The entangled ergodic theorem in the almost periodic case. Linear Algebra Appl.2010, 432, 526–535. [Google Scholar] [CrossRef]
Fidaleo, F. On strong ergodic properties of quantum dynamical systems. Infin. Dimens. Anal. Quantum Probab. Relat. Top.2009, 12, 551–564. [Google Scholar] [CrossRef]
Fidaleo, F.; Mukhamedov, F. Strict weak mixing of some C*-dynamical systems based on free shifts. J. Math. Anal. Appl.2007, 336, 180–187. [Google Scholar] [CrossRef]
Fidaleo, F.; Mukhamedov, F. Ergodic properties of Bogoliubov automorphisms in free probability. Infin. Dimens. Anal. Quantum Probab. Relat. Top.2010, 13, 393–411. [Google Scholar] [CrossRef]
Crismale, V.; Fidaleo, F.; Lu, Y.G. Ergodic theorems in quantum probability: An application to the monotone stochastic processes. Ann. Sc. Norm. Super Pisa Cl. Sci.2017, 17, 113–141. [Google Scholar]
Wiener, N.; Wintner, A. On the ergodic dynamics of almost periodic systems. Am. J. Math.1941, 63, 794–824. [Google Scholar] [CrossRef]
Anzai, H. Ergodic skew product transformations on the torus. Osaka Math. J.1951, 3, 83–99. [Google Scholar]
Sz-Nagy, N. Sur les contractions de I’espace de Hilbert. Acta Szeged1953, 15, 87–92. [Google Scholar]
Reed, M.; Simon, B. Functional Analysis; Academic Press: New York, NY, USA; London, UK, 1980. [Google Scholar]
Mukhamedov, F. On tensor products of weak mixing vector sequences and their applications to uniquely E-weak mixing C*-dynamical systems. Bull. Aust. Math. Soc.2012, 85, 46–59. [Google Scholar] [CrossRef]
Furstenberg, H. Recurrence in Ergodic Theory and Combinatorial Number Theory; Princeton University Press: Princeton, NJ, USA, 1981. [Google Scholar]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely
those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or
the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas,
methods, instructions or products referred to in the content.