Classical finite association schemes lead to a finite-dimensional algebras
which are generated by finitely many stochastic matrices. Moreover, there exist
associated finite hypergroups. The notion of classical discrete association
schemes can be easily extended to the possibly infinite case. Moreover, the
notion of association schemes can be relaxed slightly by using suitably
deformed families of stochastic matrices by skipping the integrality
conditions. This leads to larger class of examples which are again associated
to discrete hypergroups.
In this paper we propose a topological generalization of the notion of
association schemes by using a locally compact basis space X and a family of
Markov-kernels on X indexed by a further locally compact space D where the
supports of the associated probability measures satisfy some partition
property. These objects, called continuous association schemes, will be related
to hypergroup structures on D. We study some basic results for this new
notion and present several classes of examples. It turns out that for a given
commutative hypergroup the existence of an associated continuous association
scheme implies that the hypergroup has many features of a double coset
hypergroup. We in particular show that commutative hypergroups, which are
associated with commutative continuous association schemes, carry dual positive
product formulas for the characters. On the other hand, we prove some rigidity
results in particular in the compact case which say that for given spaces X,D
there are only a few continuous association schemes
