We show that all positive functionals as well as all derivations of the noncommutative Schwartz space are continuous. We also prove this space is not boundedly approximately amenable however it is approximately amenable.

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Keywords:Fréchet space Schauder basis Mean ergodic operator Power bounded operator We characterize quasi-reflexive Fréchet spaces with a basis in terms of the properties of this basis. As a consequence we prove that a Fréchet space with a basis is quasi-reflexive of order one if and only if for every power bounded operator T , either T or T is mean ergodic.

In this note we investigate amenability properties in the class of the so‐called DF‐algebras. We also characterize, in terms of the defining sequence of weights, amenable Köthe co‐echelon algebras.

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