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Spectral synthesis in multidimensional Fourier algebras

Kanupriya, N. Shravan Kumar

TL;DR

The paper extends spectral synthesis theory to the multidimensional Fourier algebra $A^n(G)$ on locally compact groups. It develops an operator-space framework with Haagerup-type tensor products, establishes functorial and support properties, and proves multidimensional subgroup lemmas and injection/inverse projection theorems for spectral and Ditkin sets, including normal and amenable-group variants. A key result is the parallel synthesis between $A^n(G)$ and $A^{n+1}(G)$ in compact groups, enabling a multidimensional analogue of Malliavin's theorem. The work culminates in a multidimensional Malliavin-type dichotomy: for abelian $G$, all closed subsets of $G^n$ are spectral iff $G$ is discrete, and in general the synthesis properties force discreteness and related structural constraints on $G$.

Abstract

Let $G$ be a locally compact group and let $A^n(G)$ denote the $n$-dimensional Fourier algebra, introduced by Todorov and Turowska. We investigate spectral synthesis properties of the multidimensional Fourier algebra $A^n(G).$ In particular, we prove versions of the subgroup lemma, injection, and inverse projection theorems for both spectral sets and Ditkin sets. Additionally, we provide a result on the parallel synthesis between $A^n(G)$ and $A^{n+1}(G)$ and finally prove Malliavin's theorem.

Spectral synthesis in multidimensional Fourier algebras

TL;DR

The paper extends spectral synthesis theory to the multidimensional Fourier algebra on locally compact groups. It develops an operator-space framework with Haagerup-type tensor products, establishes functorial and support properties, and proves multidimensional subgroup lemmas and injection/inverse projection theorems for spectral and Ditkin sets, including normal and amenable-group variants. A key result is the parallel synthesis between and in compact groups, enabling a multidimensional analogue of Malliavin's theorem. The work culminates in a multidimensional Malliavin-type dichotomy: for abelian , all closed subsets of are spectral iff is discrete, and in general the synthesis properties force discreteness and related structural constraints on .

Abstract

Let be a locally compact group and let denote the -dimensional Fourier algebra, introduced by Todorov and Turowska. We investigate spectral synthesis properties of the multidimensional Fourier algebra In particular, we prove versions of the subgroup lemma, injection, and inverse projection theorems for both spectral sets and Ditkin sets. Additionally, we provide a result on the parallel synthesis between and and finally prove Malliavin's theorem.
Paper Structure (18 sections, 34 theorems, 61 equations)

This paper contains 18 sections, 34 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Tensor product of operator spaces
  4. Multidimensional Fourier algebra
  5. Functorial properties and Support
  6. Functorial Properties
  7. Support of element of $VN^n(G)$
  8. Subgroup lemma and injection theorem for sets of spectral synthesis
  9. Spectral synthesis in commutative Banach algebras
  10. Subgroup lemma and injection theorem
  11. Ideals with bounded approximate identities
  12. Ditkin sets
  13. Subgroup lemma
  14. Injection theorem
  15. Inverse projection theorem
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $H$ is a closed subgroup of a locally compact group $G$ and $X$ an $A^n(G)$-submodule of $VN^n(G).$ Then a closed subset $E$ of $H^n$ is a set of $X_H$-synthesis for $A^n(H)$ if and only if $E$ is a set of $X$-synthesis for $A^n(G).$

Theorems & Definitions (60)

  • Theorem
  • Theorem
  • Theorem
  • Theorem
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Lemma 3.5
  • ...and 50 more