On the homological and algebraical properties of some Feichtinger algebras.

Let G be a locally compact group (not necessarily abelian) and B be a homogeneous Banach space on G, which is in a good situation with respect to a homogeneous function algebra on G. Feichtinger showed that there exists a minimal Banach space B_min in the family of all homogenous Banach spaces C on G, containing all elements of B with compact support. In this paper, the amenability and super amenability of B_min with respect to the convolution product or with respect to the pointwise product are showed to correspond to amenability, discreteness or finiteness of the group G and conversely. We also prove among other things that B_min is a symmetric Segal subalgebra of L¹(G) on an IN-group G, under certain conditions, and we apply our results to study pseudo-amenability and some other homological properties of B_min on IN-groups. Furthermore, we determine necessary and sufficient conditions on A under which A_min with the pointwise product is an abstract Segal algebra or Segal algebra in A, whenever A is a homogeneous function algebra with an approximate identity. We apply these results to study amenability of some Feichtinger algebras with respect to the pointwise product. [ABSTRACT FROM AUTHOR]