### A uniformly bounded representation associated to a free set in a discrete group

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Answering a question of Pisier, posed in [10], we construct an L-set which is not a finite union of translates of free sets.

Let $G$ be a locally compact group, for $p\in (1,\infty )$ let $P{f}_{p}\left(G\right)$ denote the closure of ${L}^{1}\left(G\right)$ in the convolution operators on ${L}^{p}\left(G\right)$. Denote ${W}_{p}\left(G\right)$ the dual of $P{f}_{p}\left(G\right)$ which is contained in the space of pointwise multipliers of the Figa-Talamanca Herz space ${A}_{p}\left(G\right)$. It is shown that on the unit sphere of ${W}_{p}\left(G\right)$ the $\sigma ({W}_{p},P{f}_{p})$ topology and the strong ${A}_{p}$-multiplier topology coincide.

We show that for the t-deformed semicircle measure, where 1/2 < t ≤ 1, the expansions of ${L}_{p}$ functions with respect to the associated orthonormal polynomials converge in norm when 3/2 < p < 3 and do not converge when 1 ≤ p < 3/2 or 3 < p. From this we conclude that natural expansions in the non-commutative ${L}_{p}$ spaces of free group factors and of free commutation relations do not converge for 1 ≤ p < 3/2 or 3 < p.

For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [, ]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the products...

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