The talk describes a surprisingly rich family of function spaces which can be defined on general LCA (locally compact Abelian groups, such as G = R^d). The start point is the Banach Gelfand triple (SO,L2,SO'), consisting of the Segal algebra SO(G) as a space of test functions and the dual space as the minimal resp. maximal space in this family. One of the most attractive (and surprising) facts about this setting, which requires only the use of Banach spaces and their dual spaces, is the existence of a kernel theorem, which extends the classical association of L2-kernels with the family of Hilbert-Schmidt operators.

As time permits a number of questions arising from classical analysis and time-frequency analysis resp. Gabor analysis are mentioned.