# 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.

**DOI: **http://dx.doi.org/10.7900/jot.2016jun27.2114

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

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