This book consists of 5 chapters: In chapter 1, the authors gave an overview of the book and its contents. They started by giving a short introduction to Riesz transforms and their \(Lp\)-boundedness in various settings in which they have been studied in the literature. They also explained the emergence of noncommutative \(Lp\)-spaces and noncommutative geometry in this context. Furthermore, they described their main results concerning Riesz transforms, functional calculus of Hodge-Dirac operators, and spectral triples. They equally presented examples that can be used with their results and ended with an overview of the contents of the other chapters. In chapter 2, the authors started by recalling the used properties of the notions used in this book: operators, semigroups, H functional calculus, noncommutative \(Lp\)-spaces, and probabilities. In particular, the construction of their Markovian semigroups of Fourier and Schur multipliers is a standing assumption in the rest of the book. They equally investigated vector-valued unbounded bilinear forms on Banach spaces which will be used as a framework for (the domain of) the carré du champ. They also showed a transference result between Fourier multipliers on the group von Neumann algebras and Fourier multipliers on the crossed product of von Neumann algebras. Then they gave useful results on Hilbertian valued noncommutative \(Lp\) spaces for the sequel of the book. Finally, they examined in detail the carré du champ and the first order differential calculus for semigroups of Fourier multipliers and semigroups of Schur multipliers. In chapter 3, the authors started by proving Khintchine inequalities for \(q\)-Gaussians in crossed products. As a consequence, they obtained boundedness of \(Lp\)-Riesz transforms associated with Markovian semigroups of Fourier multipliers and defined over these crossed products with \(q\)-Gaussians. They also gave dependence in \(p\) and independence of the group \(G\) and the Markovian semigroup, of these \(Lp\) inequalities. Then they examined in detail the domains of the operators related to Kato’s square root problem for Markovian semigroups of Fourier multipliers. They also showed how to extend the carré du champ associated with such a Markovian semigroup to a closed form. Moreover, they solved the Kato square root problem for Markovian semigroups of Schur multipliers. In particular, in its course, they again proved Khintchine inequalities for \(q\)-Gaussians. They also obtained the constants of the Kato square root problem independently of the Markovian semigroup and discuss dependence in \(p\). Finally, they also investigated Meyer’s problem for semigroups of Schur multipliers and study the \(Lp\)-boundedness of directional Riesz transforms. In chapter 4, the authors introduced Hodge-Dirac operators associated with Markovian semigroups of Fourier multipliers and they show that these operators admit a bounded H functional calculus on a bisector. They also provided Hodge decompositions. They equally showed a similar result for Hodge-Dirac operators associated with Markovian semigroups of Schur multipliers. Particular attention is paid to the domain of the Hodge-Dirac operators and different choices are discussed in the course of the chapter. They also proved the independence of the bounds of the H functional calculus, on the group \(G\) or index set \(I\), and on the Markovian semigroup of multipliers. In chapter 5, the authors started by giving an overview of quantum (locally) compact metric spaces. Then, they showed that we can associate quantum compact metric spaces to some Markov semigroups of Fourier multipliers satisfying additional conditions: injectivity and a gap condition on the cocycle which represents the semigroup, and the finite dimensionality (with explicit control on \(p\)) of the cocycle Hilbert space. They showed a similar result for semigroups of Schur multipliers and obtained a quantum locally compact metric space. They further explored the connections of our gap condition between Fourier multipliers and Schur multipliers with some examples. In the sequel, they introduced spectral triples (=noncommutative manifolds) associated with Markov semigroups of Fourier multipliers or Schur multipliers satisfying again some technical conditions, and in all, they investigate four different settings. Along the way, they introduced a Banach space variant of the notion of spectral triple suitable for their context. Finally, they investigated the bisectoriality and the functional calculus of some Hodge-Dirac operators which are crucial in the noncommutative geometries which they introduced here.