Invariant means on weakly almost periodic functions and generalized fixed point properties

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

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A Note on the Topological Group c0

Axioms ◽

10.3390/axioms7040077 ◽

2018 ◽

Vol 7 (4) ◽

pp. 77

Author(s):

Michael Megrelishvili

Keyword(s):

Banach Space ◽

Topological Group ◽

Unit Sphere ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions ◽

Semigroup Compactification ◽

Weakly Almost Periodic ◽

Open Question

A well-known result of Ferri and Galindo asserts that the topological group c 0 is not reflexively representable and the algebra WAP ( c 0 ) of weakly almost periodic functions does not separate points and closed subsets. However, it is unknown if the same remains true for a larger important algebra Tame ( c 0 ) of tame functions. Respectively, it is an open question if c 0 is representable on a Rosenthal Banach space. In the present work we show that Tame ( c 0 ) is small in a sense that the unit sphere S and 2 S cannot be separated by a tame function f ∈ Tame ( c 0 ) . As an application we show that the Gromov’s compactification of c 0 is not a semigroup compactification. We discuss some questions.

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Almost Periodic Solutions for Second Order Dynamic Equations on Time Scales

Discrete Dynamics in Nature and Society ◽

10.1155/2013/784198 ◽

2013 ◽

Vol 2013 ◽

pp. 1-11 ◽

Cited By ~ 2

Author(s):

Yan Wang ◽

Lan Li

Keyword(s):

Fixed Point ◽

Periodic Solutions ◽

Time Scales ◽

Existence And Uniqueness ◽

Second Order ◽

Almost Periodic Functions ◽

Dynamic Equations ◽

Almost Periodic Solutions ◽

Almost Periodic ◽

Periodic Functions

We firstly introduce the concept and the properties ofCmalmost periodic functions on time scales, which generalizes the concept of almost periodic functions on time scales and the concept ofC(n)-almost periodic functions. Secondly, we consider the existence and uniqueness of almost periodic solutions for second order dynamic equations on time scales by Schauder’s fixed point theorem and contracting mapping principle. At last, we obtain alternative theorems for second order dynamic equations on time scales.

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