2020/09/07 by Adrián M. González‐Pérez, González-Pérez, Adrián M.
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2009.03019
openalex publication_date 2020/09/07 · openalex created_date 2020/09/11 · openalex updated_date 2026/07/28
Let Γ\curvearrowright Ω be a measure-preserving action and L Γ\hookrightarrow L^∞(Ω) \rtimes Γ the natural inclusion of the group von Neumann algebra into the crossed product. When μ(Ω) = ∞, we have that this natural embedding is not trace-preserving and therefore does not extends boundedly to the associated noncommutative Lp-spaces. Nevertheless, we show that when Ω has an invariant mean there is an isometric embedding of Lp(L Γ) into an ultrapower of Lp(Ω\rtimes Γ) that intertwines Fourier multipliers and is L Γ-bimodular. As a consequence we obtain the lower transference bound ‖ Tm: Lp(L Γ) → Lp(L Γ) ‖ ≤ ‖ (id \rtimes Tm): Lp(Ω\rtimes Γ) → Lp(Ω\rtimes Γ) ‖, and the same follows for complete norms. The condition of having an invariant mean is quite restrictive. Therefore, we explore whether other equivariant embeddings Φ: L Γ→ L^∞(Ω) yield a more general transference result. We show that the transference proof above works verbatim whenever Φ is completely positive, amenable (in the sense of inducing an amenable correspondence) and intertwines Fourier multipliers at the L2-level. Although no new transference results are obtained, both the classification of equivariant maps and the study their amenability may be of independent interest to some readers.