2022/05/27 by Arhancet, Cédric, Kriegler, Christoph
#FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2205.13823
We study the algebra \mathfrakM∞,dec(G) of decomposable Fourier multipliers on the group von Neumann algebra VN(G) of a locally compact group G, and its relation to the Fourier-Stieltjes algebra B(G). For discrete groups, we prove that these two algebras coincide isometrically. In contrast, we show that the identity \mathfrakM∞,dec(G) = 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 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 Lp-spaces Lp(VN(G)), for 1 ≤ p ≤ ∞. 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 = ∞. These results reveal new structural links between harmonic analysis, operator algebras, and the geometry of locally compact groups.