The paper is devoted to studying the Fredholmness in the Banach algebra \({\mathcal A}(\Omega)\) generated by the discrete convolution operators acting on \(\ell^2({\mathbb Z})\) with kernels in the Fourier image \({\mathcal F}(\Omega)\) of a Banach subalgebra \(\Omega\) of \(L^\infty({\mathbb T})\). It is assumed that the coefficients \(\varphi\in\ell^\infty({\mathbb Z})\) of the convolution operators are such that the limits \(\lim_{n\to\pm\infty}| \varphi(n)| \) exist and are finite. For an arbitrary Banach subalgebra \(\Omega\subset L^\infty({\mathbb T})\), a necessary condition for the Fredholmness of an operator \(A\in{\mathcal A}(\Omega)\) is obtained in terms of its presymbol and the commutator ideal of \({\mathcal A}(\Omega)\) is described in terms of the so-called Nikolski ideals. If \(C({\mathbb T})\subset\Omega\subset QC({\mathbb T})\), where \(QC({\mathbb T})\) is the algebra of quasicontinuous functions, then the commutator ideal of \({\mathcal A}(\Omega)\) coincides with the ideal of compact operators on \(\ell^2({\mathbb T})\). In this case, a criterion for the Fredholmness of an operator \(A\in{\mathcal A}(\Omega)\) and an index formula for \(A\) are obtained.
The results of this paper are discrete analogues of those presented by K.A.Georgiev and V.M.Deundyak [St.Petersbg.Math.J.11, No.2, 269–284 (2000); translation from Algebra Anal.11, No.2, 88–108 (1999; Zbl 0945.47052)]. No proofs are given.