| Summary: | Consider a uniquely ergodic <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>*</mo> </msup> </semantics> </math> </inline-formula>-dynamical system based on a unital *-endomorphism <inline-formula> <math display="inline"> <semantics> <mo>Φ</mo> </semantics> </math> </inline-formula> of a <inline-formula> <math display="inline"> <semantics> <msup> <mi>C</mi> <mo>*</mo> </msup> </semantics> </math> </inline-formula>-algebra. We prove the uniform convergence of Cesaro averages <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> <msubsup> <mo mathsize="small">∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>λ</mi> <mrow> <mo>−</mo> <mi>n</mi> </mrow> </msup> <mo>Φ</mo> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> for all values <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula> 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.
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