Differential operators on locally compact groups (Q910869)

The author defines and discusses differential operators on locally compact topological groups on the basis of the space of \(C^{\infty}\)- functions defined by G. Czichowski, K. P. Rudolph and the reviewer. There are three chapters. In Chapter 0. Preliminaries, classical facts on the structure of locally compact groups with respect to Lie groups - one- parameter subgroups, pro-Lie groups, Lie algebras, the adjoint representation etc. - are reported. In Chapter 1. Spaces of differentiable functions, the notion of differentiability given by G. Czichowski, K. P. Rudolph and the reviewer for pro-Lie groups is extended to arbitrary locally compact groups without countability assumptions. Topologies on the spaces of differentiable functions and on its duals, the spaces of generalized functions are discussed. Evidently, the author is unaware of a paper of \textit{G. Volk} [Math. Nachr. 87, 161-170 (1979; Zbl 0334.46041)]. The last point of this chapter is devoted to the connections between left invariant continuous linear maps and generalized functions. Support decreasing linear maps on the defined spaces of differentiable functions are defined as differential operators in Chapter 2. Differential operators. The algebra of differential operators is endowed with a suitable topology making it a topological algebra. The explicit form of differential operators as sums of derivations associated to one- parameter subgroups multiplied by \(C^{\infty}\)-functions is exhibited (theorem 2.3). Moreover it is proved, that the topological algebra of generalized functions with support in the identity of the group is topologically isomorphic to the topological algebra of left invariant differential operators (theorem 2.4). There is a description of the center of the algebra of left invariant differential operators by means of the adjoint representation (theorem 2.5).