An Application of von Neumann Algebras to Finite Difference Equations

Abstract

Here we assume that P(D) is a properly elliptic differential operator (with complex coefficients), homogeneous of order 2m, and we use the convention D = i(a/ax). A Fourier transform of (1) with respect to the boundary variables formally reduces the problem to a family of boundary value problems for ordinary differential equations on a half line. If the number of boundary conditions we impose is half the order of P(D) and if we require that P(D) is properly elliptic, then it follows by a trivial dimension argument that uniqueness of solution in the homogeneous problem is equivalent to existence of solution in the inhomogeneous problem for general data {fj}. (See for example [7], Chapter II, ? 4.) In this paper we obtain an analogous result for finite difference approximations of boundary value problems on a half space. Our proof of the equivalence in the difference case is also based on a dimension argument, but it is the dimension function of a von Neumann algebra of type II, rather than the dimension function of linear algebra, which appears. This argument is an application of Breuer's theory [2] of Fredholm operators in a von Neumann algebra and is closely related to our joint paper [3]. We consider an approximation of P(D) by a difference operator