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In this article, we give a representation of bounded complex linear operators that preserve idempotent elements on the Fourier algebra of a locally compact group. When such an operator is, moreover, positive or contractive, we show that the operator is induced by either a continuous group homomorphism or a continuous group antihomomorphism. If the groups are totally disconnected, bounded homomorphisms on the Fourier algebra can be realized by the idempotent preserving operators.

Let H be an ultraspherical hypergroup and let $A(H)$ be the Fourier algebra associated with $H.$ In this paper, we study the dual and the double dual of $A(H).$ We prove among other things that the subspace of all uniformly continuous functionals on $A(H)$ forms a $C^*$-algebra. We also prove that the double dual $A(H)^{\ast \ast }$ is neither commutative nor semisimple with respect to the Arens product, unless the underlying hypergroup H is finite. Finally, we study the unit elements of $A(H)^{\ast \ast }.$

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Linear preservers on idempotents of Fourier algebras

ON THE ALGEBRAS $\boldsymbol {VN(H)}$ AND $\boldsymbol {VN(H)^{*}} $ OF AN ULTRASPHERICAL HYPERGROUP $\boldsymbol {H}$

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Linear preservers on idempotents of Fourier algebras

ON THE ALGEBRAS $\boldsymbol {VN(H)}$ AND $\boldsymbol {VN(H)^{*}} $ OF AN ULTRASPHERICAL HYPERGROUP $\boldsymbol {H}$