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Using the Godement mean M of positive-type functions over a groupG, we study “M-abelian systems” {<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V3507G3151865G8X_html\220_2005_Article_BF01651215_TeX2GIFIE1.gif" border="0" alt="$$\mathfrak{A}$$" />, a} of aC*-algebra<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V3507G3151865G8X_html\220_2005_Article_BF01651215_TeX2GIFIE2.gif" border="0" alt="$$\mathfrak{A}$$" /> and a homomorphic mapping a of a groupG into the homomorphism group of<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V3507G3151865G8X_html\220_2005_Article_BF01651215_TeX2GIFIE3.gif" border="0" alt="$$\mathfrak{A}$$" />. Consideration of the Godement mean off(g)Ug withf a positive-type function overG andU a unitary representation ofG first yields a generalized mean-ergodic theorem. We then define the Godement mean off(g) p(ag(A)) withA e<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V3507G3151865G8X_html\220_2005_Article_BF01651215_TeX2GIFIE4.gif" border="0" alt="$$\mathfrak{A}$$" /> and p a covariant representation of the system {<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V3507G3151865G8X_html\220_2005_Article_BF01651215_TeX2GIFIE5.gif" border="0" alt="$$\mathfrak{A}$$" />, a} for which theG-invariant Hilbert space vectors are cyclic and study its properties, notably in relation with ergodic and weakly mixing states over<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V3507G3151865G8X_html\220_2005_Article_BF01651215_TeX2GIFIE6.gif" border="0" alt="$$\mathfrak{A}$$" />. Finally we investigate the “discrete spectrum” of covariant representations of {<img src="/fulltext-image.asp?format=htmlnonpaginated&src=V3507G3151865G8X_html\220_2005_Article_BF01651215_TeX2GIFIE7.gif" border="0" alt="$$\mathfrak{A}$$" />, a} (i.e. the direct sum of the finite-dimensional subrepresentations of the associated representations ofG).

Ergodic states in a non commutative ergodic theory