Theorems of Bochner and Lévy for nuclear groups (Q2762062)

The author surveys extensions of theorems of Bochner, on positive definite functions, and Lévy, on sequential continuity, to nuclear groups (a class of topological Abelian groups containing locally compact Abelian groups and locally convex nuclear spaces). For example, let \(G\) denote a topological Abelian group with character group \(\Gamma\). Assume that \(\Gamma\) separates the points of \(G\) and it is equipped with the compact open topology. Then \(G\) is said to satisfy the Bochner property if every positive definite function on \(G\) equals the (pre)-Fourier transform of a (necessarily unique) Radon measure on \(\Gamma\). In Theorem 12.1 of [Additive subgroups of topological vector spaces. Lect. Notes Math. 1466 (Berlin etc. 1991; Zbl 0743.46002)] the author proved a result implying that nuclear groups have the Bochner property. The latter extends theorems of Bochner, Minlos and Weil-Raikov.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00016].