# Averaging weakly almost periodic Schur multipliers

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”.

• 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. Nov 3 '20 at 15:31
• 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. 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 $$\varphi$$.