Henrik Kreidler: A Peter-Weyl type theorem for compact transitive groupoids

A corollary of the classical Peter-Weyl theorem for compact groups asserts that if

T ∈ L (E) is a linear isometry on a Banach space E with relatively compact orbits, then

T has discrete spectrum, i.e., the eigenspaces of T span a dense subspace of E. This can

be used to obtain an operator theoretic characterization of “structured systems” in

topological dynamics: If ϕ : K → K is a homeomorphism of a metrizable compact space

K, then there is an invariant metric for ϕ if and only if the induced Koopman operator on

C(K) has discrete spectrum.

In this talk we discuss a Peter-Weyl type result replacing compact groups by compact

groupoids. Groupoids generalize groups in the sense that not every pair of elements has

to be composable. They play an important role in differential geometry, algebraic

topology and non-commutative geometry.

As an application of our result we characterize “structured extensions” of dynamical

systems opening the door to a systematic operator theoretic approach. This is joint work

with Nikolai Edeko (University of Tübingen).