The<i>C</i>^∗-algebras of some real and<i>p</i>-adic solvable groups

Abstract

When G is a locally compact group, the unitary representation theory of G is the "same" as the ^representation theory of the group C*-algebra C*(G). Hence it is of interest to determine the isomorphism class of C*(G) for a wide variety of groups G. Using methods suggested by papers of Z'ep and Delaroche, we determine explicitly the C*-algebras of the "ax + b" groups over all nondiscrete locally compact fields and of a number of two-step solvable Lie groups. Only finitely many C*-algebras arise as the group C*-algebras of 3-dimensional simply connected Lie groups, and we characterize many of them. We also discuss the C*-algebras of unipotent p-adic groups.