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Hypergroups and hypergroup algebras. (English. Russian original) Zbl 0628.43009
J. Sov. Math. 38, 1734-1761 (1987); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 26, 57-106 (1985).
This paper is a survey of the subject of hypergroup theory (or alternatively, of the theory of generalized translation operators). A very general form of the theory is concerned with the following situation. Let $$\Phi$$ be a space of functions on a set H. Suppose that for each x in H a linear operator $$R^ x$$ (called right translation by x) is given on $$\Phi$$. For $$y\in H$$ define left translation $$L^ y$$ by y using the formula $$L^ y \phi (x)=R^ x \phi (y)$$. Then we have a hypergroup if $$L^ y$$ maps $$\Phi$$ into itself for each y, each $$L^ y$$ commutes with each $$R^ x$$, and there is e in H for which $$R^ e$$ is the identity.
With this structure, to any pair y, x of points of H can be assigned as ‘generalized product’ the linear functional $$\phi \mapsto R^ x \phi (y)$$ on $$\Phi$$ ; in the case in which H is a locally compact space and $$\Phi$$ is C(H) this will often mean assigning a measure as the product of two points of H. (An alternative approach to the definition can be obtained by this route.) In the latter case the space of measures of compact support on H receives a natural multiplication known as generalized convolution. In a similar way, if H is a $$C^{\infty}$$- manifold, a hypergroup structure provides a convolution for distributions on H. Hypergroups of particular interest arise if the product of any two elements of H is a probability measure or if there is an ‘involution’ on H (to play part of the role of inverses in a group).
Being very general, hypergroups embrace a wide range of particular examples. Obviously familiar convolution algebras of measures are included. Any finite hypercomplex system (i.e. finite-dimensional associative algebra) can be viewed as a hypergroup, as can a hypercomplex system with a continuous basis. The double coset space of a locally compact group with respect to a compact subgroup has a natural hypergroup structure. Certain differential operators generate translation operators and thence hypergroups.
The theory of hypergroups aims at generalizing much of harmonic analysis. This survey considers Lie theory, Laplace and Fourier transforms, spectral synthesis, representation theory, Plancherel’s theorem and other topics. There is a list of 183 references (about half from the Russian literature) dated up to 1985.
Reviewer: J.S.Pym

##### MSC:
 43A99 Abstract harmonic analysis 58B25 Group structures and generalizations on infinite-dimensional manifolds
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