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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

Citations:

Zbl 0325.42017
Full Text: EuDML