Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability

Abstract

We study the algebra ${\mathfrak{M}}^{\infty ,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$, and its relation to the Fourier-Stieltjes algebra $\mathrm{B}(G)$. For discrete groups, we prove that these two algebras coincide isometrically. In contrast, we show that the identity ${\mathfrak{M}}^{\infty ,\mathrm{dec}}(G)=\mathrm{B}(G)$ fails for various classes of non-discrete groups, and that, among second-countable unimodular groups, inner amenability ensures the equality. Our approach relies on the existence of contractive projections preserving complete positivity from the space of completely bounded weak* continuous operators on $\mathrm{VN}(G)$ onto the subspace of completely bounded Fourier multipliers. We show that such projections exist in the inner amenable case. As an application, we obtain a new operator-algebraic characterization of amenability. We also investigate the analogous problem for the space of completely bounded Fourier multipliers on the noncommutative ${\mathrm{L}}^p$-spaces ${\mathrm{L}}^p(\mathrm{VN}(G))$, for $1\le p\le \infty$. Using Lie group theory and results stemming from the solution to Hilbert's fifth problem, we prove that second-countable unimodular finite-dimensional amenable locally compact groups admit compatible projections at $p=1$ and $p=\infty$. These results reveal new structural links between harmonic analysis, operator algebras, and the geometry of locally compact groups.

66 pages, improvements, some parts have been removed and will be integrated into a companion paper

References in corpus (5)

• Metric geometry of locally compact groups
• The Haagerup property for locally compact quantum groups
• Group C*-algebras without the completely bounded approximation property
• Schur and Fourier multipliers of an amenable group acting on non-commutative Lp-spaces
• Existence of tracial states on reduced group C*-algebras