K-théorie algébrique négative et K-théorie topologique de l'algèbre de fréchet des opérateurs régularisants

We prove that the algebra of smoothing operators with coefficients in a Fréchet algebra is K0-regular. Consequently the same result is true for Schatten ideals. They are new examples of K0-Regular Algebras of functional analysis type after those of J. Rosenberg. As a consequence, negative algebraic K-theory is equal to topological K-theory for smoothing operators and for trace operators ideal with coefficients in a Fréchet algebra stable by holomorphic functional calculus. This results extend those of M. Karoubi about Schatten ideals of dimension > 1.