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.

Freedom from equine infectious anaemia virus infection in Spanish Purebred horses

Introduction: No cases of equine infectious anaemia (EIA) have been reported in Spain since 1983. Factors that could increase the risk of reintroducing equine infectious anaemia virus (EIAV) into Spain include the recent occurrence of the disease in Europe and the absence of compulsory serological testing before importation into Spain. Aims and objectives: Given the importance of the Spanish Purebred (SP) horse breeding industry in Spain, the aim of this cross-sectional study was to provide evidence of freedom from EIAV in SP stud farms in Central Spain. Materials and methods: Serum samples from 555 SP horses, collected between September 2011 and November 2013, were tested using a commercially available EIAV ELISA with a published sensitivity of 100 per cent. Results: All 555 samples were negative for antibody to EIAV, providing evidence of a true EIAV seroprevalence between 0 per cent and 0.53 per cent (95% CIs of the sensitivity and specificity of the ELISA technique used Q10 were 100 per cent and 99.3 per cent, respectively) among the SP breeding population in Central Spain. Conclusions: These findings should serve to increase confidence when exporting SP horses to other countries.