An Algebra of Boundary Value Problems Not Requiring Shapiro–Lopatinskij Conditions

We construct an algebra of pseudo-differential boundary value problems that contains the classical Shapiro–Lopatinskij elliptic problems as well as all differential elliptic problems of Dirac type with APS boundary conditions, together with their parametrices. Global pseudo-differential projections on the boundary are used to define ellipticity and to show the Fredholm property in suitable scales of spaces.