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Functional and harmonic analysis of noncommutative Lp spaces associated to compact quantum groups

Abstract : This thesis is devoted to studying the analysis on compact quantum groups. It consists of two parts. First part presents the classification of invariant quantum Markov semigroups on these quantum homogeneous spaces. The generators of these semigroups are viewed as Laplace operators on these spaces.The classical sphere, the free sphere, and the half-liberated sphere are considered as examples and the generators of Markov semigroups on these spheres are classified. We compute spectral dimensions for the three families of spheres based on the asymptotic behavior of the eigenvalues of their Laplace operator.In the second part, we study of convergence of Fourier series for non-abelian groups and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as some summation methods and mean convergence of associated noncommutative Fourier series. We establish a general criterion of maximal inequalities for approximative identities of noncommutative Fourier multipliers. As a result, we prove that for any countable discrete amenable group, there exists a sequence of finitely supported positive definite functions, so that the associated Fourier multipliers on noncommutative Lp-spaces satisfy the pointwise convergence. Our results also apply to the almost everywhere convergence of Fourier series of Lp-functions on non-abelian compact groups. On the other hand, we obtain the dimension free bounds of noncommutative Hardy-Littlewood maximal inequalities in the operator-valued Lp space associated with convex bodies.
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Xumin Wang. Functional and harmonic analysis of noncommutative Lp spaces associated to compact quantum groups. Functional Analysis [math.FA]. Université Bourgogne Franche-Comté, 2019. English. ⟨NNT : 2019UBFCD040⟩. ⟨tel-02549788⟩

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