Continuous invariant averagings

Main results: For every equicontinuous almost periodic linear representation of a group in a complete locally convex space L with the countability property, there exists the unique invariant averaging; it is continuous and is expressed by using the L-valued invariant mean of Bochner and von-Neumann. An analog of Wiener s approximation theorem for an equicontinuous almost periodic linear representation in a locally convex space with the countability property is proved.