An operator theoretical approach to enveloping Psi*- 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) t hat the small Melrose algebra A := Psi(b, cl)( 0)(X, (b)Omega(1/2)) (cf. [2 2], [23]) of classical, totally characteristic pseudodifferential operators carries no topology such that it is a topological algebra with an open gro up of invertible elements, in particular, the algebra A cannot be spectrall y 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 sub algebra of L (rho(b) L-2 (X, (b)Omega(1/2))) by a generalization of a metho d of GOHBERG and KRUPNIK. Furthermore, A is densely embedded in a Frechet a lgebra A subset of or equal to B which is a Psi*-algebra in the sense of GR AMSCH [9, Definition 5.1], reflecting also smooth properties of the origina l algebra A.