HGFei, Oct. 2017

The theory of Segal algebras is going back to the research done by Hans Reiter, partially found in his book(s) and lecture notes.

From a modern view-point they are dense Banach ideals in L1(G), over a locally compact group.

The ideal theorem tells us that there is a natural bijection between the closed ideals of L1 and those of a Segal algebra S, simply by taking the intersection of a closed ideal I of L1 with S, or otherwise (inverse mapping) taking the closer of a closed ideal J in S in the L1-norm.

This fact can be used to explain, that the theory of spectral analysis, originally formulated in the L1-setting, can equivalently be discussed in the setting of any given Segal algebra

Nowadays the prototypical choice would be the Segal algebra SO(G).