Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability

TL;DR

This work develops an operator-algebraic bridge between the Fourier-Stieltjes algebra $B(G)$ and the space of decomposable Fourier multipliers on the group von Neumann algebra $VN(G)$. For discrete groups, the authors show an isometric identification of $B(G)$ with the decomposable multiplier space, while for many non-discrete groups the inclusion is strict; inner amenability yields equality in second-countable unimodular cases, linking amenability with complete positivity-preserving projections. A central technical feat is constructing contractive projections from the space of completely bounded weak* continuous operators on $VN(G)$ onto the multiplier subspace, enabling a new characterization of amenability in operator-algebraic terms. The paper also extends the analysis to noncommutative $L^p$-spaces and proves compatible projections at $p=1$ and $p= obreak obreak ∞$ for second-countable unimodular finite-dimensional amenable groups, illuminating deep connections among harmonic analysis, group geometry, and operator algebra theory.

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 $1 \leq p \leq \infty$. 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.

Theorems & Definitions (77)

• Theorem 1.1
• Definition 1.2
• Theorem 1.3: Lau-Paterson
• Conjecture 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 1.8
• Theorem 1.9
• Theorem 1.10