Abstract
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.
Highlights
T, the classical ergodic theory primarily deals with the long time behavior of the Cesaro means
Among the most famous classical ergodic theorems, we mention the Birkhoff individual ergodic theorem concerning the study of the point-wise limit limn→+∞ Mn ( f )( x ) and the von Neumann mean ergodic theorem concerning the limit L2 − limn→+∞ Mn ( f ), whenever f is square-summable
We mention several unconventional ergodic theorems (e.g., [2]), which play a fundamental role in number theory
Summary
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. 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 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]). We end the paper with some example based on the tensor product, which is nontrivial, of an Anzai skew product (cf. [17]) and a uniquely mixing noncommutative dynamical system, for which ph ph the sequence Ma,λ (n) n∈N does not converge for some a ∈ A and λ ∈ σpp (Vφ,Φ )\σpp (Φ)
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