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. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 770 – 780 ¨ ITAK ˙ c TUB ⃝ doi: Continuous invariant averagings Djavvat KHADJIEV,∗Abdullah C ¸ AVUS ¸ Department of Mathematics, Karadeniz Technical University, Trabzon, 61080, Turkey Received: • Accepted: • Published Online: • Printed: Abstract: 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. Key words: Invariant averaging, invariant mean, almost periodic function 1. Introduction The concept of the invariant averaging was introduced and investigated in the paper [14]. The following results were obtained in it: (i). A group G is amenable if and only if every almost periodic linear representation of G in a quasi-complete locally convex space has an invariant averaging; (ii). A locally compact group G is compact if and only if every strongly continuous linear representation of G in a quasi-complete locally convex space has an invariant averaging. The invariant averaging can be considered as an infinite-dimensional analog of Reynold’s operator in the invariant theory [21, ]. Invariant averagings are closely connected with vectorvalued invariant means, amenable groups, almost periodic functions, almost periodic linear representations of a group in locally convex spaces and uniformly equicontinuous actions of a group on compacts. The vector-valued invariant mean with values in a locally convex space was introduced and applied to investigation of vector-valued almost .
