Resumen de Noncommutative analysis techniques in the geometry of Lp spaces and Calderón-Zygmund theory

Antonio Ismael Cano Marmol

• The contents of this dissertation can be encompassed within the area of noncommutative harmonic analysis. A central feature of this research field is the substitution of functions defined over measure spaces by operators acting on Hilbert spaces.

  In the first part of this thesis, we study the applicability of the theory of functions on the Hamming cube to the geometry of Banach spaces and Banach space embedding theory. In particular, our work finds it inspiration in a result by Naor and Schechtman, known as metric Xp inequalities, about the nonembeddability as a metric space of the Lebesgue space Lq(0,1) into Lp(0,1) whenever 2 < q < p. We extend these results to the context of group von Neumann algebras and noncommutative Lp spaces.

  The second part of this dissertation is concerned with the extension of the theory of Calderón-Zygmund operators to the context of operator-valued functions. We give an in-depth description about the semicommutative space of bounded mean oscillation operators and the semicommutative Hardy space. Moreover, we study the boundedness of singular integrals with noncommuting kernels from the column Hardy space to the semicommutative L1 space.