Positive isometric Fourier multipliers on non-commutative $L^p$-spaces

TL;DR

This paper characterizes positive isometric Fourier multipliers on non-commutative L^p-spaces associated with group von Neumann algebras, showing they correspond to continuous characters of the group for p ≠ 2.

Contribution

It extends the characterization of positive isometric Fourier multipliers to non-unimodular groups, linking them to continuous characters in the non-commutative L^p setting.

Findings

Positive isometric Fourier multipliers are characterized by continuous characters for p ≠ 2.

The result generalizes previous unimodular group cases.

The characterization applies to all locally compact groups, not just special classes.

Abstract

For a locally compact group \(G\), let \(\mathcal{L}G\) denote its left group von Neumann algebra and let \(L^p(\mathcal{L}G)\), \(1 \le p \le \infty\), be the corresponding non-commutative \(L^p\)-space. Given \(\phi \in L^\infty(G)\), we study the Fourier multiplier \(M_{\phi,p}\) acting on \(L^p(\mathcal{L}G)\). We prove that for any \(p \neq 2\), the operator \(M_{\phi,p}\) is a positive surjective isometry if and only if \(\phi\) coincides locally almost everywhere with a continuous character of \(G\). This characterization extends results obtained recently (jointly with C.~Arhancet) in the unimodular setting.