A Noncommutative Wiener Lemma and A Faithful Tracial State on Banach Algebras of Time-Frequency Shift Operators

TL;DR

This paper establishes a noncommutative Wiener lemma and constructs a faithful tracial state on a Banach algebra of time-frequency shifts, leading to insights on linear independence in time-frequency analysis.

Contribution

It introduces a noncommutative Wiener lemma and constructs a faithful tracial state on the algebra of time-frequency shifts, advancing understanding of its structure.

Findings

Proves a noncommutative Wiener lemma for the algebra.

Constructs a faithful tracial state on the algebra.

Provides a partial proof of the Heil-Ramanathan-Topiwala conjecture.

Abstract

In this paper we analyze the Banach *-algebra of time-frequency shifts with absolutely summable coefficients. We prove a noncommutative version of the Wiener lemma. We also construct a faithful tracial state on this algebra which implies the algebra contains no compact operators. As a corollary we obtain a special case of the Heil-Ramanathan-Topiwala conjecture regarding linear independence of finitely many time-frequency shifts of one $L^2$ function.