Mackey topology on locally convex spaces and on locally quasi-convex groups. Similarities and historical remarks

A counterpart of the Mackey–Arens Theorem for the class of locally quasi-convex topological Abelian groups (LQC-groups) was initiated in Chasco et al. (Stud Math 132(3):257–284, 1999). Several authors have been interested in the problems posed there and have done clarifying contributions, although the main question of that source remains open. Some differences between the Mackey Theory for locally convex spaces and for locally quasi-convex groups, stem from the following fact: The supremum of all compatible locally quasi-convex topologies for a topological abelian group G may not coincide with the topology of uniform convergence on the weak quasi-convex compact subsets of the dual groupG∧. Thus, a substantial part of the classical Mackey–Arens Theorem cannot be generalized to LQC-groups. Furthermore, the mentioned fact gives rise to a grading in the property of “being a Mackey group”, as defined and thoroughly studied in Díaz Nieto and Martín-Peinador (Proceedings in Mathematics and Statistics 80:119–144, 2014). At present it is not known—and this is the main open question—if the supremum of all the compatible locally quasi-convex topologies on a topological group is in fact a compatible topology. In the present paper we do a sort of historical review on the Mackey Theory, and we compare it in the two settings of locally convex spaces and of locally quasi-convex groups. We point out some general questions which are still open, under the name of Problems.

The making and unmaking of heritage

Heritage is defined by history which is by nature multi layered. The passage of time and the perspectives it affords, enables and even necessitates constant reexamination and reinterpretation of history. What effect do changes in historical perspective then have upon the definition of heritage which relies on an understanding of its history? The present paper attempts to engage with the notion of heritage, criteria of its definition, and the mutable nature of such designations with specific reference to architectural constructions and historical cities that enjoy or have enjoyed in the past the status of a ‘World Heritage Site’. Examples such as the Louvre museum in Paris or the King’s Cross station in London make an interesting study as they not only allow insight into the past but reflect the changes and adaptation over a period of time. Multiple alterations, some very recently, have modified them extensively since the time they were accorded the ‘World Heritage Site’ status. The above examples are contrasted by sites ridden with conflict such as the Bamiyan Valley. This site has been placed under the ‘World Heritage In Danger’ list by UNESCO taking into account the destruction of the Buddha statues in the region. The act of vandalism itself has had dual implications. While causing an irreparable loss to mankind of its heritage, it also serves as an effective symbol of religious fanaticism that is a pressing concern of our times. The paper then moves on to explore the case of Dresden which lost its ‘World Heritage’ status with the construction of the Waldschlösschen Bridge. This is a particularly interesting case because with the absolute destruction of the city during the Second World War, it was necessary to reconstruct the historical city while simultaneously acknowledging and addressing the modern day requirements. During the reconstruction, with the readaptation of the spaces, it was almost impossible to replicate the original architectural program or to undertake such a large reconstruction project employing only the traditional techniques and materials. This essentially made it a new city constructed in the image of the old. The recent necessity of a growing city was met by the construction of a bridge that has caused it to lose its ‘World Heritage’ status. Finally, this paper endeavours to foster discussion of questions central to the definition of heritage such as what happens when we have to adapt a living space to avoid its deterioration and descent into dereliction by overuse. Does it necessarily lose its historical value? What exactly is Historical value?.