The C^∗-algebras of some real and p-adic solvable groups

Jonathan Rosenberg

Vol. 65 (1976), No. 1, 175–192

DOI: 10.2140/pjm.1976.65.175

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.

Mathematical Subject Classification 2000

Primary: 22D25

Secondary: 22E25

Milestones

Received: 2 December 1975

Published: 1 July 1976