$$C^*$$-algebras, $$H^*$$-algebras and trace ideals of pseudo-differential operators on locally compact, Hausdorff and abelian groups

Abstract

We define pseudo-differential operators on a locally compact, Hausdorff and abelian group G as natural extensions of pseudo-differential operators on $${\mathbb {R}}^n$$. In particular, for pseudo-differential operators with symbols in $$L^2(G\times \widehat{G})$$, where $$\widehat{G}$$ is the dual group of G, we give explicit formulas for the products and adjoints, characterize them as Hilbert–Schmidt operators on $$L^2(G)$$ and prove that they form a $$C^*$$-algebra, which is also a $$H^*$$-algebra. We give a characterization of trace class pseudo-differential operators in terms of symbols lying in a subspace of $$L^1(G\times \widehat{G})\cap L^2(G\times \widehat{G})$$.