Let $\Gamma$ be a countable (discrete) group and let $\varphi:\Gamma\times\Gamma\to\mathbb{C}$ be a (non-equivariant) Schur multiplier. See Chapter 5 of [2] for details. Assume that, for all $t\in\Gamma$, the function \begin{align*} s\longmapsto\varphi(st,s) \end{align*} is weakly almost periodic. Let $m$ be the unique invariant mean on WAP$(\Gamma)$; see Section 3 of [1].

Question: Is it true that the function $\psi:\Gamma\to\mathbb{C}$ given by \begin{align*} \psi(t)=m(s\mapsto\varphi(st,s)) \end{align*} is a Herz-Schur multiplier on $\Gamma$? Can we estimate the norm of $\psi$ in terms of the norm of $\varphi$?

First naive attempt: By Theorem 5.1 in [2], there is a Hilbert space $H$ and bounded functions $\xi, \eta:\Gamma\to H$ such that \begin{align*} \varphi(t,s)=\langle\xi(s),\eta(t)\rangle,\quad\forall s,t\in\Gamma. \end{align*} This allows us to write \begin{align*} \psi(s^{-1}t)=m\left(r\mapsto\langle\xi(rs),\eta(rt)\rangle\right),\quad\forall s,t\in\Gamma, \end{align*} but I don't know if this can be expressed as a scalar product on a suitable Hilbert space.

[1] Uffe Haagerup, Søren Knudby, and Tim de Laat. A complete characterization of connected Lie groups with the approximation property. Ann. Sci. Éc. Norm. Supér. (4), 49(4):927-946, 2016.

[2] Gilles Pisier. Similarity problems and completely bounded maps, volume 1618 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, expanded edition, 2001. Includes the solution to “The Halmos problem”.

  • 2
    $\begingroup$ My answer below is fine, but to go more in your direction: the map $m(r \mapsto \langle \xi_r, \eta_r\rangle)$ is a scalar product on $\ell_\infty(\Gamma;H)$. So the completion with respect to the semi-norm $\|(\xi_r)_r\|^2:= m(r \mapsto \|\xi_r\|_H^2)$ is the Hilbert space you were looking for. $\endgroup$ Nov 3 '20 at 15:31
  • $\begingroup$ A clarification : for my comment above to make sense, one needs first to extend $m$ to a mean on $\ell_\infty(\Gamma)$. This is possible by Hahn-Banach. $\endgroup$ Nov 4 '20 at 14:18

Ignacio, the answer is yes. Indeed, there is a net $m_i$ of probability measures on $G$ such that $m(f) = \lim_i \int f dm_i$ for every $f \in \textrm{WAP}(\Gamma)$. One justification of this is as follows: extend $m$ to a (perhaps non-invariant) mean on $\ell_\infty(\Gamma)$, and use the standard weak-* density of $\ell_1(\Gamma)$ in its bidual. See page 1 of Greenleaf's Invariant Means on Topological Groups and their Applications.

The answer to your question follows directly: we have $\psi(s^{-1}t) = \lim_i \int \varphi(rs,rt) dm_i(r)$, so $(s,t) \mapsto \psi(s^{-1}t)$ is a pointwise limit of a net of Schur multipliers with norm at most the norm of $\varphi$, so is a Schur multiplier with norm at most the norm of $\varphi$.


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