On the existence and structure of psi*-algebras of totally characteristic operators on compact manifolds with boundary

As a contribution to the pseudodifferential analysis on manifolds with sing ularity we construct for each smooth. compact manifold X with boundary a Ps i*-algebra A infinity(b)(X, (b)Ohm(1/2)) subset of or equal to L(rho(b)L(2) (X, (b)Ohm(1/2))) containing the algebra Psi(b, cl)(0)(X, (b)Ohm(1/2)) tota lly characteristic pseudodifferential operators introduced by Melrose [25] in 1981 as a dense subalgebra; further, there is a homomorphism tau(A)((b)) :(X, (b)Ohm(1/2)) --> Q(Psi)((b)) characterizing the Fredholm property of a epsilon A(infinity)((b))(X, (b)Ohm(1/2)) by means of the invertibility of tau(A)((b)) (a) epsilon Q(Psi)((b)), where Q(Psi)((b)) is an algebra of C-i nfinity-symbols reflecting the smooth structure of the manifold X. The Fred holm inverses of Fredholm operators in A(infinity)((b))(X, (b)Q(1/2)) are a gain in the algebra A(infinity)((b))(X, (r)Q(1/2)) and we have elliptic reg ularity corresponding to the scale rho(b)H(b)(m)(X, (b)Ohm(1/2)) of b-Sobol ev spaces naturally associated to X. Localized to the interior of X we reco ver the ordinary pseudodifferential calculus. Finally, spectrum, Jacobson t opology and the relationship of certain closed ideals in the algebra A(infi nity)((b))(X, (b)Ohm(1/2)) are described explicitly. (C) 1999 Academic Pres s.