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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.

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A definition of an invariant averaging for a linear representation of a group in a locally convex space is given. Main results: A group $H$ is finite if and only if every linear representation of $H$ in a locally convex space has an invariant averaging. A group $H$ is amenable if and only if every almost periodic representation of $H$ in a quasi-complete locally convex space has an invariant averaging. A locally compact group $H$ is compact if and only if every strongly continuous linear representation of $H$ in a quasi-complete locally convex space has an invariant averaging.

A formula is set up between vector-valued mean and scalar-valued means that enables us translate many important results about scalar-valued means developed in [1] to vector-valued means. As applications of the theory of vector-valued means, we show that the definitions of a mean in [2] and [3] are equivalent and the space of vector-valued weakly almost periodic functions is admissible.

Means, generalized means and invariant means (on a semigroup) with values
in a Banach lattice are defined and studied.

We show mean ergodic theorems for vector-valued weakly almost periodic functions (in the sense of Eberlein) defined on a semigroup which take values in a locally convex topological vector space. Next, motivated by Frechet [10], we study the relationship between almost periodicity of semigroups of mappings and their equicontinuity, and also prove mean ergodic theorems for equicontinuous semigroups.

We show that a Banach space X is complemented in its ultraproducts if and only if for every amenable semigroup S the space of bounded X-valued functions defined on S admits (a) an invariant average; or (b) what we shall call “an admissible assignment”. Condition (b) still provides an equivalence for quasi-Banach spaces, while condition (a) necessarily implies that the space is locally convex.

The purpose of this paper is to introduce and study the
notion of a vector-valued $\pi\text{-invariant}$ mean
associated to a unitary representation $\pi$ of a locally
compact group $G$ on $\mathcal{S},$ a self-adjoint linear
subspace containing $I$ of $\mathcal{B}(H_\pi).$ We obtain,
among other results, an extension theorem for
$\pi\text{-invariant}$ completely positive maps and
$\pi\text{-invariant}$ means which characterizes amenability
of $G.$ We also study vector-valued means on $\mathcal{S}$ of
$\pi\text{-(weakly)}$ almost periodic operators on $H_\pi.$

An approach to a definition of an integral, which differs from definitions of Lebesgue and Henstock–Kurzweil integrals, is considered. We use trigonometrical polynomials instead of simple functions. Let V be the space of all complex trigonometrical polynomials without the free term. The definition of a continuous integral on the space V is introduced. All continuous integrals are described in terms of norms on V. The existence of the widest continuous integral is proved, the explicit form of its norm is obtained and it is proved that this norm is equivalent to the Alexiewicz norm. It is shown that the widest continuous integral is wider than the Lebesgue integral. An analog of the fundamental theorem of calculus for the widest continuous integral is given.

Properties of diverse classes of almost periodic functions with values in locally convex spaces and of almost periodic representations on locally convex spaces are considered. The well-known criterion for the almost periodicity of weakly almost periodic group representations on Banach spaces (in terms of scalar almost periodicity) is extended to the case of weakly continuous weakly almost periodic representations on barrelled spaces in which the weakly closed convex hulls of weakly compact sets are weakly compact. Applications of this result are indicated and a survey of the current state of some other classical problems in the theory of almost periodic functions (as applied to almost periodic functions with values in locally convex spaces) and modern directions of investigation related to almost periodic functions on groups and finite-dimensional unitary representations of groups are presented. In particular, decomposition problems for weakly almost periodic representations and characterizations of diverse classes of almost periodic functions (including criteria for almost periodicity), existence problems for the mean value, countability conditions for the spectrum of a scalarly almost periodic function, theorems on the integral and the differences of almost periodic functions, and other relationships among strong, scalar, and weak almost periodicity for functions with values in locally convex spaces are treated.