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Isomorphisms of hypergroups. (English) Zbl 0776.43001
The algebraic structure of a locally compact hypergroup $K$ as introduced by Dunkl, Jewett and Spector is defined via the convolution on the Banach space $M\left(K\right)$ of all bounded measures. The purpose of the paper is to extend the isomorphism theorems of Kawada and Wendel and of Johnson and Strichartz to the hypergroup setting. In particular, the following main results are proved for hypergroups ${K}_{1}$ and ${K}_{2}$ with Haar measures: Each isometric isomorphism between ${L}^{1}\left({K}_{1}\right)$ and ${L}^{1}\left({K}_{2}\right)$ can be extended to an isometric isomorphism between $M\left({K}_{1}\right)$ and $M\left({K}_{2}\right)$. Conversely, the restriction of an isometric isomorphism between $M\left({K}_{1}\right)$ and $M\left({K}_{2}\right)$ to ${L}^{1}\left({K}_{1}\right)$ leads to an isometric isomorphism between ${L}^{1}\left({K}_{1}\right)$ and ${L}^{1}\left({K}_{2}\right)$. Moreover, each isometric isomorphism between $M\left({K}_{1}\right)$ and $M\left({K}_{2}\right)$ is related to a hypergroup isomorphism between the spaces ${K}_{1}$ and ${K}_{2}$ together with some multiplicative function $\gamma \in C\left({K}_{1}\right)$.

##### MSC:
 43A62 Hypergroups (abstract harmonic analysis) 43A10 Measure algebras on groups, semigroups, etc. 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A20 ${L}^{1}$-algebras on groups, semigroups, etc.