The C∗-algebra of singular integral operators with semi-almost periodic coefficients

We establish a Fredholm criterion for the operators belonging to the C∗-algebra generated by singular integral operators with semi-almost periodic matrix coefficients. The result is applied to Toeplitz-like operators that are perturbed by integral operators with fixed singularities at infinity, in which case it leads to an effectively verifiable Fredholm criterion together with an index formula. Our approach is based on an isomorphism theorem for C∗-algebras associated with C∗-dynamical systems and the notion of canonical generalized AP factorization for almost periodic matrix functions.