Aspects of Calder\'on-Zygmund theory for von Neumann algebras I

In a series of two papers, we develop a Calder\'on-Zygmund theory which includes classical and noncommutative measure spaces (von Neumann algebras) and explore applications in abstract harmonic analysis. Algebraic tools from geometric group theory are used to study smooth Fourier multipliers in noncommutative duals of discrete groups. Our main result is a cocycle generalization of H\"ormander-Mihlin multiplier theorem, which provides new examples of Fourier multipliers even for Rn and Tn. Noncommutative Riesz transforms, Littlewood-Paley estimates, radial Fourier multipliers for arbitrary cocycles or new estimates for Schur multipliers are also investigated. Our results rely on intrinsic BMO spaces associated with a semigroup of Fourier multipliers -sometimes also called Herz-Schur multipliers- and twisted forms of semicommutative Calder\'on-Zygmund operators. Other examples include the free group algebra and noncommutative tori.