On Continuous Hochschild comology and Cohomology Groups

Abstract

Using the method of continuous projective resolutions established by Alain Connes, we calculate the continuous Hochschild homology and cohomology groups of the Frechet algebra of smooth functions on a manifold M. These (co)homology groups are allowed to take values in the Frechet space E of smooth sections of a vector bundle E over M and in the strong dual E*. Moreover, we show that in the cohomological case the differential and continuous Hochschild cohomologies are naturally isomorphic.