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 Fréchet \(*\)-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.