Fourier-Stieltjes algebras, decomposable Fourier multipliers and amenability (2205.13823v5)

Abstract: We prove that the Fourier-Stieltjes algebra $\mathrm{B}(G)$ of a discrete group $G$ is isometrically isomorphic to the algebra $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$. In contrast, we show that $\mathfrak{M}^{\infty,\mathrm{dec}}(G) \neq \mathrm{B}(G)$ for some classes of non-discrete locally compact groups, while we prove that $\mathfrak{M}^{\infty,\mathrm{dec}}(G) = \mathrm{B}(G)$ holds for any (second countable unimodular) inner amenable locally compact group. To prove these results, we leverage groupoid theory and investigate the problem of whether a contractive projection exists, preserving complete positivity, from the space of normal completely bounded operators on $\mathrm{VN}(G)$ onto the space $\mathfrak{M}^{\infty,\mathrm{cb}}(G)$ of completely bounded Fourier multipliers. We provide an affirmative solution in the inner amenable case and demonstrate that such projections do not exist for (second countable) non-amenable connected locally compact groups. Further, we investigate whether the space $\mathfrak{M}^{p,\mathrm{cb}}(G)$ of completely bounded Fourier multipliers on the noncommutative $\mathrm{L}^p$-space $\mathrm{L}^p(\mathrm{VN}(G))$ is complemented in the space of completely bounded operators, where $1 \leq p \leq \infty$. Using doubling metrics on Lie groups and structural results from the solution to Hilbert's fifth problem, we establish that any (second countable unimodular) amenable locally compact group admits compatible bounded projections at the levels $p=1$ and $p=\infty$, which has applications to decomposable Fourier multipliers. Moreover, we present a new characterization of amenability for (second countable unimodular) locally compact groups.