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The amenability of the substitution group of formal power series. (English. Russian original) Zbl 1228.43001

Izv. Math. 75, No. 2, 239-252 (2011); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 75, No. 2, 19-34 (2011).
Summary: We study the amenability property for the group \({\mathcal J}({\mathbf k})\) of formal power series in one variable with coefficients in a commutative ring \({\mathbf k}\) with identity. We show that there exists an invariant mean on the space \( C_{\text{u}}^*({\mathcal J}({\mathbf k}))\) of uniformly continuous bounded functions on this group. This is equivalent to the fact that every continuous action of \({\mathcal J}({\mathbf k})\) on every compact space has an invariant probability measure.

MSC:

43A07 Means on groups, semigroups, etc.; amenable groups
20E18 Limits, profinite groups
22A10 Analysis on general topological groups
46E15 Banach spaces of continuous, differentiable or analytic functions
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