Continuous Hochschild Cohomology and Formality

We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories. We define the corresponding Hochschild complex controlling deformations and prove formality theorems for the Fr\'echet algebras of smooth functions on a manifold, the de Rham algebra and for the Dolbeault algebra of a complex manifold. In the latter case, the Hochschild cohomology is equivalent to Kontsevich's extended deformation complex, the Hochschild cohomology of the derived category in case $X$ is a smooth projective variety and to Gualtieri's deformation complex of $X$ viewed as generalized complex manifold. We also compute the continuous Hochschild cohomology for various categories of matrix factorisations.