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Structure and analysis on nuclear groups. (English) Zbl 0978.22001

Nuclear groups were introduced by W. Banaszczyk [Additive subgroups of topological vector spaces. Lect. Notes Math. 1466 (Berlin etc. 1991; Zbl 0743.46002)] and they form a class of topological Abelian groups which contains locally compact groups and the additive groups of nuclear vector spaces and is closed under Cartesian products, subgroups and separated quotients. Even though Banaszczyk’s definition of nuclear groups is technically involved, it made possible to extend to this wide class of topological groups many results that were known to hold for nuclear spaces. Thus, the class of nuclear groups provides a good setting where to deal with basic questions related to the theory of topological Abelian groups: duality, weak topologies, etc. As a consequence, nuclear groups have attracted the attention of many researchers interested in topological groups since their introduction in 1991. Nonetheless, given the technical difficulties that one has to face in order to treat nuclear groups, a main achievement here ought to be the clarification of the structure of these groups since it would make possible a better understanding of the whole theory. In the first part of the paper under review, the author considers this question and he proves a representation theorem which permits him to describe nuclear groups as dense subgroups of projective limits of the form \(\varprojlim E_{i}/K_{i}\times D_{i}\), \(E_{i}/K_{i}\) being the quotient of a Hilbert space by a closed subgroup and \(D_{i}\) a discrete group, for all \(i\in I\). The projective limit satisfies the condition that the mappings \(T_{ij}:E_{j}/K_{j}\times D_{j}\rightarrow E_{i}/K_{i}\times D_{i}\) defining this limit are induced by linear mappings \(\overline T_{ij}:E_{j}\rightarrow E_{i}\), defined between the corresponding Hilbert spaces, which are nuclear in the usual way and commute with the canonical projections. As a consequence of the result above, the author is able to prove that each unbounded subset of a nuclear group \(G\) contains an infinite interpolation subset in the sense of Hartman and Ryll-Nardzewski; that is, a subset which is discrete and \(C^{\ast}\)-embedded in the group \(G\) equipped with the Bohr topology. This nice result extends to nuclear groups a fact first proved for discrete groups by E. van Douwen [Topology Appl. 34, 69-91 (1990; Zbl 0696.22003)]. On the other hand, the existence of interpolation sets is applied to deduce that nuclear groups strongly respect compactness in the sense of W. W. Comfort, F. J. Trigos-Arrieta and T. S. Wu [Fundam. Math. 143, 119-136 (1993; Zbl 0812.22001)] generalizing previous results given by J. Galindo and S. Hernández [Fundam. Math. 159, 195-218 (1999; Zbl 0934.22008)]. Finally, the last part of the paper is dedicated to the study of several duality properties of nuclear groups. Here, the main finding accomplished by the author is that each complete nuclear group coincides algebraically with its bidual group. The paper also contains several other results of independent interest.

MSC:

22A10 Analysis on general topological groups
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
54H11 Topological groups (topological aspects)
43A40 Character groups and dual objects
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