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This book on recent research in noncommutative harmonic analysis treats the Lp boundedness of Riesz transforms associated with Markovian semigroups of either Fourier multipliers on non-abelian groups or Schur multipliers. The detailed study of these objects is then continued with a proof of the boundedness of the holomorphic functional calculus for Hodge–Dirac operators, thereby answering a question of Junge, Mei and Parcet, and presenting a new functional analytic approach which makes it possible to further explore the connection with noncommutative geometry. These Lp operations are then shown to yield new examples of quantum compact metric spaces and spectral triples. The theory described in this book has at its foundation one of the great discoveries in analysis of the twentieth century: the continuity of the Hilbert and Riesz transforms on Lp. In the works of Lust-Piquard (1998) and Junge, Mei and Parcet (2018), it became apparent that these Lp operations can be formulated on Lp spaces associated with groups. Continuing these lines of research, the book provides a self-contained introduction to the requisite noncommutative background. Covering an active and exciting topic which has numerous connections with recent developments in noncommutative harmonic analysis, the book will be of interest both to experts in no-commutative Lp spaces and analysts interested in the construction of Riesz transforms and Hodge–Dirac operators.

The thesis is concerned with the smooth functional calculus for operators with spectrum in the positive reals, more specifically spectral multiplier theorems. We start with abstract and optimal functional calculi, that is, homomorphisms u : C(K) -+ B(X). If X is a Hilbert space, then the natural operator valued extension C(K; lu]') -+ B(X) is again bounded. Using R-boundedness, a strengthening of uniform boundedness of operators, we extend this result to general Banach spaces X and apply it to the H infinity calculus and to unconditional bases in LP spaces. We develop calculi which are associated with sectorial operators. The classical examples are the spectral theorems of Mih1in and Hôrmander giving classes of smooth functions which are Fourier multipliers on LV. These theorems have alrèâdy been extended to a large class of Laplace type operators. We add a unifying theme using operator theory: we compare the Mihlin and Hôrmander calculus with the boundedness of classical operator families associated with the sectorial operator. For the family of imaginary powers, we give a characterization of their polynomial norm growth in terms of a functional calculus which refines the Mihlin calculus. We study diffusion semigroups acting on a scale of Banach spaces. If this scale is the classical LP spaces, it is known iliat the semigroup has an analytic extension on a sector in the complex plane. We give a generalization of this result to non-commutative LP-spaces using the theory of operator spaces.