Invariance of compactness for the Bohr topology

We define the g-extension of a topological Abelian group G as the set of all characters on Ĝ such that the restriction to every equicontinuous subset of Ĝ is continuous with respect to the pointwise convergence topology. A g-group is a topological Abelian group (G,τ) such that its g-extension coincides with its completion. The Bohr topology of a topological group (G,τ) is the topology that the group inherits as a subset of its Bohr compactification. A topological group (G,τ) respects a property P if the subsets A of G that satisfy the property P are exactly the same for the Bohr topology and for the original topology of the group [Trigos-Arrieta, J. Pure Appl. Algebra 70 (1991) 199]. All groups here are assumed to be Abelian. We prove that every complete g-group when endowed with its Bohr topology is a μ-space. As a consequence, we obtain that for a complete g-group the properties of respecting functionally boundedness, pseudocompactness, countable compactness and compactness are all equivalent and a characterization of this property is also provided. Finally, we extend a theorem of Rosenthal about the existence of sequences equivalent to the ℓ1-basis. We prove that for a Čech-complete g-group the property of respecting compactness is equivalent to the existence of conveniently placed sequences equivalent to the ℓ1-basis.