Decomposing positive representations on L^{p}-spaces

Abstract:

Let G be a locally compact group acting on a measure space (X,mu), where the measure mu is G-invariant. Then G acts unitarily on L^{2}(X,mu) and conditions are known that allow one to decompose the action of G into irreducible representations. Less is known about the representations of G on L^{p}(X,mu) for p not equal to 2. In this case G acts as a group of isometric lattice isomorphisms, and one question is whether this action can be decomposed into band-irreducible representations, representations for which the only invariant order closed ideals are trivial.  For decomposing unitary representations on Hilbert spaces an important tool is that of a direct integral. To replace this notion in the setting of Banach spaces we consider Banach bundles. The space of p-integrable sections of such bundles can be viewed as a p-integral of Banach spaces. Using the above techniques it will be shown that, under appropriate assumptions on X and G, the representation of G on L^{p}(X,mu) can indeed be decomposed into band-irreducible representations.

This is joint work with Marcel de Jeu from Leiden, the Netherlands.