Non-commutative harmonic analysis associated with generalized translation operators was developed within two parallel research directions, the theory of DJS-hypergroup [cf. W. R. Bloom and H. Heyer, Harmonic analysis of probability measures on hypergroups. de Gruyter Studies in Mathematics. 20. Berlin: de Gruyter (1995; Zbl 0828.43005)] and the theory of unimodular locally compact hypercomplex systems [Yu. M. Berezansky and A. A. Kalyuzhnyi, Harmonic analysis in hypercomplex systems. Mathematics and its Applications (Dordrecht). 434. Dordrecht: Kluwer (1998; Zbl 0894.43007)].
The authors introduce the notion of a locally compact hypergroup generalizing both of the above structures. Within the new framework, they study natural function spaces, Hilbert algebras, positive definite functions, von Neumann algebras generated by left and right regular representations and prove an analogue of the Pontryagin duality theorem.