An Operator Theoretical Approach to Enveloping ϕ* – and C* – Algebras of Melrose Algebras of Totally Characteristic Pseudodifferential Operators

Let X be a compact manifold with boundary. It will be shown (Theorem 3.4) that the small Melrose algebra A? ?_b,cl (χ,^bΩ^1/2) (cf. 22], 23]) of classical, totally characteristic pseudodifferential operators carries no topology such that it is a topological algebra with an open group of invertible elements, in particular, the algebra A cannot be spectrally invariant in any C* – algebra. On the other hand, the symbolic structure of A can be extended continuously to the C* – algebra B generated by A as a subalgebra of ζ(σ^bL²(χ, ^bΩ^1/2)) by a generalization of a method of Gohberg and Krupnik. Furthermore, A is densely embedded in a Fréchet algebra A ? B which is a ?* – algebra in the sense of Gramsch 9, Definition 5.1], reflecting also smooth properties of the original algebra A.