Universiteit Leiden

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Group representations in Banach spaces and Banach lattices

Promotor: S.M. Verduyn Lunel, Co-promotores: M.F.E. de Jeu, B. de Pagter

M.R. Wortel
18 april 2012
Thesis in Leiden Repository

In this thesis, group representations in Banach spaces and Banach lattices are studied. In the first part, chapter 2, a Banach algebra crossed product is constructed, which is an object that allows the translation of group representations in Banach spaces into Banach algebra representations. This can be used to understand these group representations, provided that these Banach algebra representations are sufficiently well understood. In the second part, positive group representations on Riesz spaces are studied. In chapter 3, the theory of such representations of finite groups is studied, mainly in finite dimensional spaces. The main results are that irreducible positive representations of finite groups on sufficiently nice Riesz spaces are finite dimensional, and the finite dimensional positive representations are equal to permutation representations conjugated by a diagonal matrix. In chapter 4 this is taken to the next level by considering positive representations of compact groups on Banach lattices. It turns out that these representations are often equal to an isometric positive representation conjugated by a central lattice automorphism.