C*-algebras of almost periodic pseudo-differential operators

Abstract

Our basic goal is to develop an index theory for almost periodic pseudo-differential operators on R ~. The prototype of this theory is [5] which has direct application to the almost periodic Toeplitz operators. Here, we s tudy index theory for a C*-algebra of operators on R ~ which contains most almost periodic pseudo-differential operators such as those arising in the s tudy of elliptic boundary value problems for constant coefficient elliptic operators on a half space with almost periodic boundary conditions. Our program is as follows: We begin with a discussion of a C*-algebra with symbol which contains all of the classical pseudo-differential operators on R ~. Precisely, if A is a bounded operator on L2(R ~) and 2 ER ~, let e~(A) denote the conjugate of A with the function e *~'x acting as a multiplier denoted e~. We first s tudy the C*-algebra of those A for which the function 2~+e~(A) has a strongly continuous extension to the radial compactification of R ~. The restriction of this function to the complement of R ~ then gives the usual (principal) symbol a(A) when A is a pseudo-differential operator of order zero (of a suitable type). We characterize the Fourier multipliers in this algebra and the image of the symbol map. We give sufficient conditions for the usual construction of a pseudo-differential operator as well as one of Friedrichs' constructions to give an element of this algebra. In particular, the lat ter gives a positive linear right inverse for the symbol m a p a t least when the symbol is sufficiently smooth. I n fact, we show in w 3 tha t the Friedrichs map is a right inverse to the symbol map in the almost periodic case. We expect this to be true in the general case also.