## Generalizations of the Noncommutative Grothendieck Inequality

##### Summary

In this thesis we will introduce and prove several inequalities on Banach function spaces (such as C(S) or Lp -spaces), most notably the Grothendieck inequality and the Khintchine inequality. In particular, we will study how the Khintchine inequality can be used to extend the Grothendieck inequality to other spaces.
Using some theory on C*-algebras and von Neumann algebras, we introduce the notion of noncommutative spaces that extend the definition of the usual Lp -spaces and study how the Grothendieck and Khintchine inequalities can be extended to these spaces. Finally, we will introduce arbitrary noncommutative Banach function spaces and show that if the Khintchine inequality holds for these spaces, then the Grothendieck inequality must also hold. We conclude the thesis, by introducing the concepts of concave and convex Banach function spaces and use some recent results on such spaces to state and prove a more general Grothendieck inequality.