Double orbits of weakly almost periodic functions

TL;DR

This paper investigates properties of weakly almost periodic functions on certain groups, establishing conditions under which double orbits are relatively weakly compact and exploring implications for specific group classes.

Contribution

It demonstrates that for discrete FC-groups that are WS-groups, the center has finite index, and studies groups where double orbits' compactness implies norm compactness.

Findings

Moore-groups are WS-groups.

Discrete FC-groups as WS-groups have finite index centers.

Examples include M(n) and SL(n,R) with specific orbit properties.

Abstract

A locally compact group G is called a WS-group if the double orbits of the weakly almost periodic functions on G are relatively weakly compact. It is known that Moore-groups are WS-groups. We will show that if a discrete FC-group is a WS-group then its center is of finite index in G. We will study noncompact locally compact groups with the property that if the double orbits of bounded continuous functions on G are relatively weakly compact then they are relatively norm compact. Examples of such groups include the motion group M(n) and the special linear group SL(n,R).