The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators

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 proves 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. We also estimate the coefficient decay of the inverse of finite linear combinations of time-frequency shifts.