GP-nuclear groups (Q2762067)

The classical Riemann-Dirichlet theorem states that a series of real numbers is unconditionally convergent if and only if it is absolutely convergent. The present paper analyzes how this theorem may be generalized to wider contexts, mostly focusing around the Grothendieck-Pietsch property and the concept of GP-nuclear group previously introduced by the authors [Acta Math. Hung. 88, 301-322 (2000; Zbl 0958.54042)]. A topological group \(G\) is called GP-nuclear if the groups of presummable sequences \(\ell^1\{G\}\) and absolutely summable sequences \(\ell^1(G)\) coincide algebraically and topologically. NEWLINENEWLINENEWLINEThe paper has an expository inclination, it contains well-known theorems, useful remarks and recent results often due to the authors. Although proofs of most of the central main theorems are not provided, some results are fully proved to complete the existing proof, to reveal a new angle or in order to indicate how that precise result fits into the general scheme. NEWLINENEWLINENEWLINEThe paper begins with the Dvoretzky-Rogers theorem that shows that infinite-dimensional Banach spaces always contain summable sequences that are not absolutely summable, then proceeds by introducing nuclear and summable operators to deduce that nuclearity is the key concept regarding absolute summability of summable sequences. The Grothendieck-Pietsch theorem by which the whole paper (and the notion of GP-nuclear group) is inspired is proved at this point. The proof emphasizes that nothing is lost in the classical proof if one replaces locally convex vector spaces by locally convex vector groups (a locally convex vector group differs from a locally convex vector space in that multiplication by scalars is not jointly continuous). The final section analyzes the case of nuclear groups introduced by \textit{W. Banaszczyk} [Additive subgroups of topological vector spaces. Lect. Notes Math. 1466 (Berlin etc. 1991; Zbl 0743.46002)] as a good frame for commutative harmonic analysis that encompasses nuclear locally convex spaces and locally compact Abelian groups. It is proved in that section that nuclear groups are GP-nuclear. Since the definition of GP-nuclear group is completely internal and easy to extend to other contexts (it is not for instance purely commutative) it would be interesting to know whether nuclear groups are the only (locally quasi-convex, say) GP-nuclear groups. This last question is posed and left open at the end of the paper.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00016].