2023/09/03 by Ferrer, María V., Hernández-Arzusa, Julio, Hernández, Salvador · 1 citation
#22A05 (22A25 22E99 20C15 20K20 54H11 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.2309.01223
Let G be the group of all \ZZ-valued homomorphisms of the Baer-Specker group \ZZ^\NN. The group G is algebraically isomorphic to \ZZ(\NN), the infinite direct sum of the group of integers, and equipped with the topology of pointwise convergence on \ZZ^\NN, becomes a non reflexive prodiscrete group. It was an open question to find its dual group G. Here, we answer this question by proving that G is topologically isomorphic to \ZZ^\NN⊗Q\TT, the (locally quasi-convex) tensor product of \ZZ^\NN and \TT. Furthermore, we investigate the reflexivity properties of the groups of Cp(X,\ZZ), the group of all \ZZ-valued continuous functions on X equipped with the pointwise convergence topology, and Ap(X), the free abelian group on a 0-dimensional space X equipped with the topology tp(C(X,\ZZ)) of pointwise convergence topology on C(X,\ZZ). In particular, we prove that Ap(X)≃ Cp(X,\ZZ)⊗Q\TT and we establish the existence of 0-dimensional spaces X such that Cp(X,\ZZ) is Pontryagin reflexive.