Amenable hypergroups. (English) Zbl 0755.43003

A hypergroup is a locally compact space \(K\) whose finite Borel measures have a convolution (*) structure preserving positivity and so that \(\mu\to\int_ Kd\mu\) is a multiplicative functional. One can define left translation by \(\delta_ x\) (unit point mass at \(x \in K\)) for functions in \(L^ \infty(K)\). A left invariant mean \(m\) is a functional so that \(m(\delta_ x*f) = m(f)\) for all \(x \in K\), \(f \in L^ \infty(K)\). The convex set of such means is called \(\text{LIM}(L^ \infty(K))\). The hypergroup is called amenable if this set is nonempty. The major topic of this paper is to study the effect on amenability of operations such as forming subhypergroups, quotients, automorphism orbits, and so on. Compact, commutative, and central hypergroups are amenable. If \(G\) is an amenable locally compact topological group, \(B\) is a subgroup of the automorphism group \(\text{Aut}(G)\) whose closure is compact in \(\text{Aut}(G)\), then \(G_ B\) (the space of \(B\) orbits in \(G\)) is an amenable hypergroup.
The join of a compact hypergroup and a discrete amenable hypergroup is amenable (a “join” is the construction of “fattening” the identity element of a discrete hypergroup to a compact hypergroup). There is a necessary and sufficient condition for amenability which is based on Reiter’s condition for locally compact groups (related to certain approximate identities for \(L^ 1\)). Finally there are results on the cardinalities of the set \(\text{LIM}(L^ \infty(K))\) and other sets of means.


43A62 Harmonic analysis on hypergroups
43A07 Means on groups, semigroups, etc.; amenable groups
43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc.
46H05 General theory of topological algebras