On extending characters

Abstract

Abstract. Amplifying modestly on recent work of R. W. Henrichs, we indicate that, while the impossibility of extending a continuous character uV on a subgroup H of a topological group G to a character on G is often due to algebraic problems, the impossibility of extending uV to a continuous positive definite function on G can often be viewed as a problem with uniform continuity.Recently R. W. Henrichs [3] proved the following attractive results. Theorem 1. Let G be a connected locally compact group which b not the direct product of a vector group and a compact group. Then there is a continuous character on a closed subgroup of G which cannot be extended to a continuous positive definite function on G. (ii) Characters of closed subgroups always have continuous positive definite extensions.(iii) G is the semidirect product of a vector group V and a compact group K such that K acts on V effectively as a finite group. These theorems and our treatment of an example on the affine group of the line in [7] prompted us to investigate the possibility that some characters on the subgroups (as in the theorems) might not extend to uniformly continuous functions on the whole group. (We remind the reader that continuous positive definite functions on topological groups are uniformly continuous [5, Theorem 32.4].)We note first, that it will often be impossible to extend a character of a subgroup to a character of the containing group. For example, such nonextendable characters will always exist on a normal abelian subgroup 77 of a group G if 77 is not contained in the center of G; thus, if 77 is a cyclic subgroup of order 3 of the