[Paper Review] $L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups

Matthew Wiersma|arXiv (Cornell University)|Sep 9, 2014

Advanced Operator Algebra Research20 references3 citations

TL;DR

This paper introduces and studies $L^p$-Fourier and Fourier-Stieltjes algebras $A_{L^p}(G)$ and $B_{L^p}(G)$ for locally compact groups $G$, generalizing classical Fourier algebras via $L^p$-matrix coefficients of unitary representations. The key contribution is a complete characterization of Fourier-Stieltjes ideals of $SL(2,\mathbb{R})$ as precisely $B_{L^p}(G)$ for $p \in [2,\infty)$, showing these algebras are distinct for each $p$ and form a complete invariant for the group.

ABSTRACT

Let $G$ be a locally compact group and $1\leq p

Motivation & Objective

• To extend the theory of Fourier algebras by introducing $L^p$-Fourier and Fourier-Stieltjes algebras for locally compact groups.
• To investigate how structural properties of $A_{L^p}(G)$ and $B_{L^p}(G)$ reflect the underlying group $G$.
• To characterize the Fourier-Stieltjes ideals of $SL(2,\mathbb{R})$ using $L^p$-representations.
• To demonstrate that $B_{L^p}(G)$ are distinct for each $p \in [2,\infty)$ on $SL(2,\mathbb{R})$.

Proposed method

• Define $L^p$-representations as unitary representations where matrix coefficients lie in $L^p(G)$ for a dense set of vectors in the Hilbert space.
• Construct $A_{L^p}(G)$ as the space of matrix coefficients of $L^p$-representations and $B_{L^p}(G)$ as its weak*-closure in $B(G)$.
• Use the Fell topology and weak containment properties of unitary representations to analyze the structure of $B_{L^p}(G)$.
• Apply the Cowling-Haagerup-Howe theorem to show that mock discrete series and principal series are $L^{2+\epsilon}$-representations.
• Leverage Pukánszky's tensor product theorem on complementary series representations to relate weak containment to $p$-dependence.
• Use Fell’s absorption principle and the structure of induced representations to classify ideals in $B(G)$.

Experimental results

Research questions

• RQ1 Which locally compact groups admit distinct $L^p$-Fourier-Stieltjes algebras for different $p \in [2,\infty)$?
• RQ2 Can the Fourier-Stieltjes ideals of $SL(2,\mathbb{R})$ be fully characterized in terms of $L^p$-representations?
• RQ3 For which $p$ is the complementary series representation $\pi_r$ weakly contained in an $L^p$-representation of $SL(2,\mathbb{R})$?
• RQ4 Are $B_{L^p}(G)$ and $C^*_{L^p}(G)$ complete invariants for $SL(2,\mathbb{R})$?
• RQ5 Do non-amenable or non-compact groups exhibit richer structures in $L^p$-Fourier-Stieltjes algebras than classical Fourier algebras?

Key findings

• The $L^p$-Fourier-Stieltjes algebras $B_{L^p}(SL(2,\mathbb{R}))$ are distinct for every $p \in [2,\infty)$.
• The $L^p$-Fourier algebra $A_{L^p}(G)$ is a complete invariant for $SL(2,\mathbb{R})$, meaning $A_{L^p}(G) = A_{L^q}(G)$ if and only if $p = q$.
• The complementary series representation $\pi_r$ is weakly contained in an $L^p$-representation if and only if $r \in [2/p - 1, 0)$.
• All nontrivial Fourier-Stieltjes ideals of $SL(2,\mathbb{R})$ are either the full algebra $B(G)$ or $B_{L^p}(G)$ for some $p \in [2,\infty)$.
• The $C^*$-algebras $C^*_{L^p}(SL(2,\mathbb{R}))$ are distinct for each $p \in [2,\infty)$, extending the result of Okayasu to $SL(2,\mathbb{R})$.
• The free group $\mathbb{F}_\infty$ admits a continuum of Fourier-Stieltjes ideals not of the form $B_{L^p}(\mathbb{F}_\infty)$, showing the characterization is specific to $SL(2,\mathbb{R})$.