Homomorphisms with small bound between Fourier algebras

Abstract

Inspired by Kalton and Wood’s work on group algebras, we describe almost completely contractive algebra homomorphisms from Fourier algebras into Fourier–Stieltjes algebras (endowed with their canonical operator space structure). We also prove that two locally compact groups are isomorphic if and only if there exists an algebra isomorphism T between the associated Fourier algebras (resp. Fourier–Stieltjes algebras) with completely bounded norm \({\left\| T \right\|_{cb}} < \sqrt {3/2} \left( {{\text{resp}}{\text{.}}{{\left\| T \right\|}_{cb}} < \sqrt {5/2} } \right)\) . We show similar results involving the norm distortion ‖T‖‖T −1‖ with universal but non-explicit bound. Our results subsume Walter’s well-known structural theorems and also Lau’s theorem on the second conjugate of Fourier algebras.

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Correspondence to Jean Roydor.

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Kuznetsova, Y., Roydor, J. Homomorphisms with small bound between Fourier algebras. Isr. J. Math. 217, 283–301 (2017). https://doi.org/10.1007/s11856-017-1446-6

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