A global theory of algebras of generalized functions

We present a geometric approach to defining an algebra (G) over cap (M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space L'(M) of distributions on M. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of (G) over cap (M). (G) over cap (M) is a differential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of L'(M) into (G) over cap (M) that renders l(infinity)(M) a faithful subalgebra of (G) over cap (M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus (G) over cap (M) retains all the distinguishing properties of the local theory in a global context. (C) 2002 Elsevier Science (USA).