Dissertation
Translation invariant Banach function spaces on groups
When a physical system possesses symmetries, these are often accompanied by a representation of the symmetry group on a Hilbert space. Group representations on Hilbert spaces have been extensively studied.
- Author
- C. Ding
- Date
- 19 June 2025
- Links
- Thesis in Leiden Repository
In such situations, however, there are also natural representations of the symmetry group on Banach lattices. These are functional-analytic structures in which concepts such as “positive” and “greater than” have meaning—unlike in Hilbert spaces, where ordering plays no role. The representation theory of groups on Banach lattices is still in its early stages. This thesis studies a natural class of such representations, namely those of a group on Banach lattices of functions defined on the group itself. One can, for example, think of the Banach lattice of all integrable functions on the circle group, on which the circle group acts via rotations. More generally, there is a representation of a group on any Banach lattice of functions on that group that is invariant under translations corresponding to the group operations.
The first theme addresses a number of fundamental questions about the structure of such representations. When can every function in a translation-invariant Banach function space be approximated arbitrarily closely by continuous functions that vanish outside a compact set? When by functions on which the group acts continuously and that vanish at infinity? Under mild conditions, both turn out to be equivalent to the so-called order-continuity of the norm on the space. The second theme classifies positive operators between two translation-invariant Banach function spaces that commute with left translations. These turn out to be precisely the right-convolutions with positive measures on the group.