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Polynomial growth and functional calculus in algebras of integrable cross-sections

Felipe I. Flores

TL;DR

The paper develops a unified framework for L^1-algebras of Fell bundles over groups with polynomial growth, introducing a weighted dense subalgebra L^{1,ν}(G|C) and a smooth functional calculus for self-adjoint cross-sections. A key innovation is the strengthened Young inequality that links L^p norms to operator norms on the Hilbert C*-module, enabling polynomial-growth bounds for e^{itΦ} and a Dixmier–Baillet calculus. Under symmetry, the authors prove the Wiener property, local regularity, and $^*$-regularity, and they establish norm-controlled inversion on a dense symmetric subalgebra in the discrete case. These results deepen the spectral and structural understanding of cross-section algebras and have direct implications for convolution algebras arising from C*-dynamical systems.

Abstract

Let ${\sf G}$ be a locally compact group with polynomial growth of order $d$, a polynomial weight $ν$ on ${\sf G}$ and a Fell bundle $\mathscr C\overset{q}{\to}{\sf G}$. We study the Banach $^*$-algebras $L^1({\sf G}\,\vert\,\mathscr C)$ and $L^{1,ν}({\sf G}\,\vert\,\mathscr C)$, consisting of integrable cross-sections with respect to ${\rm d} x$ and $ν(x){\rm d} x$, respectively. By exploring new relations between the $L^p$-norms and the norm of the Hilbert $C^*$-module $L^2_{\rm e}({\sf G}\,\vert\,\mathscr C)$, we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy $$\|{e^{itΦ}}\|=O(|t|^n),\quad\text{ as }|t|\to\infty,$$ for values of $n$ that only depend on $d$ and the weight $ν$. We use this fact to develop a smooth functional calculus for such elements. We also give some sufficient conditions for these algebras to be symmetric. As consequences, we show that these algebras are locally regular, $^*$-regular and have the Wiener property (when symmetric), among other results. Our results are already new for convolution algebras associated with $C^*$-dynamical systems.

Polynomial growth and functional calculus in algebras of integrable cross-sections

TL;DR

The paper develops a unified framework for L^1-algebras of Fell bundles over groups with polynomial growth, introducing a weighted dense subalgebra L^{1,ν}(G|C) and a smooth functional calculus for self-adjoint cross-sections. A key innovation is the strengthened Young inequality that links L^p norms to operator norms on the Hilbert C*-module, enabling polynomial-growth bounds for e^{itΦ} and a Dixmier–Baillet calculus. Under symmetry, the authors prove the Wiener property, local regularity, and -regularity, and they establish norm-controlled inversion on a dense symmetric subalgebra in the discrete case. These results deepen the spectral and structural understanding of cross-section algebras and have direct implications for convolution algebras arising from C*-dynamical systems.

Abstract

Let be a locally compact group with polynomial growth of order , a polynomial weight on and a Fell bundle . We study the Banach -algebras and , consisting of integrable cross-sections with respect to and , respectively. By exploring new relations between the -norms and the norm of the Hilbert -module , we are able to show that the growth of the self-adjoint, compactly supported, continuous cross-sections is polynomial. More precisely, they satisfy for values of that only depend on and the weight . We use this fact to develop a smooth functional calculus for such elements. We also give some sufficient conditions for these algebras to be symmetric. As consequences, we show that these algebras are locally regular, -regular and have the Wiener property (when symmetric), among other results. Our results are already new for convolution algebras associated with -dynamical systems.
Paper Structure (10 sections, 32 theorems, 138 equations)

This paper contains 10 sections, 32 theorems, 138 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Integrable cross-sections of polynomial growth
  4. A dense symmetric subalgebra
  5. Some consequences
  6. Preservation of spectra and --regularity
  7. Minimal ideals of a given hull
  8. The Wiener property
  9. Norm-controlled inversion
  10. Appendix: Some computations of spectral radii

Key Result

Theorem 1.1

Let ${\sf G}$ be a locally compact group with polynomial growth of order $d$. Let $\mathfrak D$ denote either $L^{1}({\sf G}\,\vert\,\mathscr C)$ or $L^{1,\nu}({\sf G}\,\vert\,\mathscr C)$, provided that there exists a polynomial weight $\nu$ on ${\sf G}$ such that $\nu^{-1}$ belongs to $L^p({\sf G} In particular, for any complex function $f\in C_{\rm c}^\infty(\mathbb{R})$, with Fourier transform

Theorems & Definitions (86)

  • Theorem 1.1
  • Lemma 1.2: Strengthened Young's inequality
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Theorem 2.5
  • proof
  • ...and 76 more