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Journal of Operator Theory

Volume 78, Issue 1,  Summer  2017  pp. 227-243.

Isomorphisms and gap theorems for Figa-Talamanca-Herz algebras

Authors:  Jean Roydor
Author institution: Institut de Mathematiques de Bordeaux, Universite de Bordeaux, 351 Cours de la Liberation, 33405 Talence Cedex, France

Summary:  It is an open question whether the Figa-Talamanca--Herz algebra $\mathcal{A}_p(G)$ determines the group $G$. We consider Figa-Talamanca--Herz algebras equipped with their $p$-operator space structure and we prove that two locally compact groups $G$ and $H$ are isomorphic if and only if there exists an algebra isomorphism $\Phi : \mathcal{A}_p(G) \to \mathcal{A}_p(H)$ with $p$-completely bounded norm $\| \Phi \|_\mathrm{pcb} < (2^{p-2}+1/2)^{1/p}$ if $1< p\leqslant 2$ or $\| \Phi \|_\mathrm{pcb} \le (2^{1-p}+1)^{1/p}$ if $2 \leqslant p < \infty$. In our second theorem, we prove an `almost norm one'' version of Host's idempotents theorem for uniformly smooth or uniformly convex Banach spaces. As applications, we obtain several gap results: for instance for norms of idempotent $p$-completely bounded multipliers and amenability constant of Figa-Talamanca--Herz algebras.

Keywords:  Figa-Talamanca--Herz algebra, $p$-operator space, $p$-completely bounded map, uniformly smooth and uniformly convex Banach space

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