The concept of Banach Gelfand Triples arose from considerations in Gabor Analysis, an important (and practically relevant) branch of time-frequency analysis (TFA). In order to discuss questions of continuity and the mapping properties of the Gabor frame operator for general $L^2$ windows the Segal algebra $S_0$ plays an important role, together with its dual, the space $S_0'$. Together with the Hilbert space $L^2$ they form a so-called Banach Gelfand Triple.

We will show, how to properly define isomorphism and homomorphism in this category of objects (norm continuity at all three layers combined with w*/w*-continuity at the outer level) and how, for example the classical Fourier transform can be discribed using these concepts.

On the other hand we will show that Gabor analysis is naturally leading to frames, which are in fact frames with respect to the canonical Banach Gelfand Triple, consisting of $(l^1,l^2,l^infty)$.