Overview
The $0$-calculus on a manifold with boundary is a micro-localization of the Lie algebra of vector fields that vanish at the boundary. It has been used by Mazzeo, Melrose to study the Laplacian of a conformally compact metric. We give a complete characterization of those $0$-pseudodifferential operators that are Fredholm between appropriate weighted Sobolev spaces, and describe $C^{*}$-algebras that are generated by $0$-pseudodifferential operators. An important step is understanding the so-called reduced normal operator, or, almost equivalently, the infinite dimensional irreducible representations of $0$-pseudodifferential operators. Since the $0$-calculus itself is not closed under holomorphic functional calculus, we construct submultiplicative Frechet $*$-algebras that contain and share many properties with the $0$-calculus, and are stable under holomorphic functional calculus ($\Psi^{*}$-algebras in the sense of Gramsch). There are relations to elliptic boundary value problems.