On the Existence and Structure of Ψ*-Algebras of Totally Characteristic Operators on Compact Manifolds with Boundary

As a contribution to the pseudodifferential analysis on manifolds with singularities we construct for each smooth, compact manifold X with boundary a Ψ*-algebra A(b)∞(X, bΩ1/2)⊆L(ϱbL2(X, bΩ1/2)) containing the algebra Ψ0b, cl(X, bΩ1/2) of totally characteristic pseudodifferential operators introduced by Melrose [25] in 1981 as a dense subalgebra; further, there is a homomorphism τ(b)A: A(b)∞(X, bΩ1/2)→Q(b)Ψ characterizing the Fredholm property of a∈A(b)∞(X, bΩ1/2) by means of the invertibility of τ(b)A(a)∈Q(b)Ψ, where Q(b)Ψ is an algebra of C∞-symbols reflecting the smooth structure of the manifold X. The Fredholm inverses of Fredholm operators in A(b)∞(X, bΩ1/2) are again in the algebra A(b)∞(X, rΩ1/2), and we have elliptic regularity corresponding to the scale ϱbHmb(X, bΩ1/2) of b-Sobolev spaces naturally associated to X. Localized to the interior of X we recover the ordinary pseudodifferential calculus. Finally, spectrum, Jacobson topology and the relationship of certain closed ideals in the algebra A(b)∞(X, bΩ1/2) are described explicitly.