An asymmetric Ellis theorem (Q2466893)

A result due to Robert Ellis states that a group with a locally compact Hausdorff topology \(\mathcal T\) making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem however does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. In the present paper the authors first establish the following related bitopological result: Let \((X,{\mathcal T},{\mathcal T}^*)\) be a pairwise Hausdorff \(k\)-bispace with locally compact symmetrization \({\mathcal T}\vee {\mathcal T}^*.\) If \((X,\cdot,{\mathcal T})\) and \((X,\cdot,{\mathcal T}^*)\) are both semitopological groups, then they are both paratopological groups and \({\mathcal T}^*={\mathcal T}^{-1}\), so that the inversion is a homeomorphism between \((X,{\mathcal T})\) and \((X,{\mathcal T}^*).\) They then introduce for a given topology \({\mathcal T}\) on a set \(X\) a new topology on \(X\) called the \(k\)-dual \({\mathcal T}^k\) of \({\mathcal T}.\) That \(k\)-dual \({\mathcal T}^k\) is defined with the help of the better known de Groot dual \({\mathcal T}^g\) of \({\mathcal T}.\) The concept of a \(k\)-dual is used to define a generalization of local compact Hausdorff which the authors call locally skew compact. In this way they obtain the following asymmetric Ellis theorem which applies to the example given above: If \((X,\cdot,{\mathcal T})\) is a locally skew compact semitopological group, then \((X,\cdot,{\mathcal T})\) and \((X,\cdot,{\mathcal T}^k)\) are both paratopological groups, and \({\mathcal T}^k={\mathcal T}^{-1},\) so that inversion is a homeomorphism between \((X,{\mathcal T})\) and \((X,{\mathcal T}^k).\) The result generalizes the classical Ellis theorem, because \({\mathcal T}={\mathcal T}^k\) whenever \((X,{\mathcal T})\) is locally compact and Hausdorff.