## Weakly almost periodic functions on hypergroups.(English)Zbl 0532.43005

TUM, Inst. Math. TUM-M8301, 12 P. (1983).
For (weakly) almost periodic complex-valued functions on hypergroups K analogues to and deviations from the classical case K a (semi)group are discussed. A hypergroup (K,*) is a locally compact space K together with an abstract multiplication * defined on $$M(K):=\{regular$$ complex-valued Borel measures on $$K\}$$ such that M(K) is a complex associative algebra with properties similar to those of the usual convolution algebra M(G) in case K a locally compact (semi)group [see e.g. R. I. Jewett, Adv. Math. 18, 1-101 (1975; Zbl 0325.42017)].
Let $$C(K):=\{f:K\to complex\quad numbers\quad | f\quad bounded\quad continuous\},$$ with sup-norm; translate $$(L_ xf)(y):=(R_ yf)(x):=f(x*y):=\int fd(\delta_ x*\delta_ y), \delta_ a:=$$ Dirac measure in a; orbit $$O_ Lf:=\{L_ xf:\quad x\in K\};$$ almost periodic functions $$AP_ L:=AP_ L(K):=\{f\in C(K):O_ Lf\quad relatively\quad compact\quad in\quad C(K)\};$$ weakly a.p. functions $WAP_ L := \left\{ f\in C(K) : O_ L f \text{ relatively weakly compact in }C(K) \text{ (and }L_ xf, R_ xf \text{ norm continuous?)}\right\}.$ Then $$AP_ L(K)$$ and $$WAP_ L(K)$$ are closed linear ”-”-invariant subspaces of C(K), one has Maak’s theorem $$AP_ L=AP_ R=:AP, WAP_ L=WAP_ R=:WAP;\quad WAP(K)=C(K)$$ iff K compact, here explicitly an $$f\in C(K)$$ not weakly a.p. is constructed; with this it is shown that (W)AP are in general not algebras under pointwise multiplication. For commutative hypergroups there exists an invariant mean on WAP.
Finally, (weak) almost periodicity on $$G^ B$$ is discussed, $$G^ B$$ the hypergroup of $$\bar B$$-orbits associated with a locally compact group G and a relatively Birkhoff-compact subgroup B of Aut(G), with $$G=R^ 2$$ and $$B=SO(2)$$ as special case.
Reviewer: H.Günzler

### MSC:

 43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions 46E15 Banach spaces of continuous, differentiable or analytic functions 22D40 Ergodic theory on groups

Zbl 0325.42017
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