A Characterization of Minimal Homogeneous Banach Spaces

Let be a locally compact group. It is shown that for a homogeneous Banach space on satisfying a slight additional condition there exists a minimal space <!-- MATH ${B_{\min }}$ --> in the family of all homogeneous Banach spaces which contain all elements of with compact support. Two characterizations of <!-- MATH ${B_{\min }}$ --> are given, the first one in terms of "atomic" representations. The equivalence of these two characterizations is derived by means of certain (bounded) partitions of unity which are of interest for themselves.