THE NONCOMMUTATIVE RESIDUE FOR MANIFOLDS WITH BOUNDARY

Let M be a compact manifold with (smooth) boundary partial derivative M of dimension greater than or equal to 2. Let B be the Boutet de Monv el algebra; it consists of couples of classical pseudodifferential sym bols with integral order having the transmission property and ''bounda ry symbols''. It can be viewed as a generalization of the algebra of c lassical pseudodifferential symbols with integral order on a compact m anifold (without boundary) to the case of manifolds with boundary. We construct a trace on B called ''the noncommutative residue'' and we sh ow that it is the only continuous trace on B lip to a multiplicative f actor. This result uses and generalizes Wodzicki's result for the case of manifolds without boundary.