  The classical as well as non commutative Korovkin-type theorems deal with
convergence of positive linear maps with respect to modes of convergences such
as norm convergence and weak operator convergence. In this article,
Korovkin-type theorems are proved for convergence of completely positive maps
with respect to weak, strong and uniform clustering of sequences of matrices of
growing order. Such modes of convergence were originally considered for
Toeplitz matrices (see [23],[26]). As an application, we translate the
Korovkin-type approach used in the finite dimensional case, in the setting of
preconditioning large linear systems with Toeplitz structure, into the infinite
dimensional context of operators acting on separable Hilbert spaces. The
asymptotic of these pre-conditioners are obtained and analyzed using the
concept of completely positive maps. It is observed that any two limit points
of the same sequence of pre-conditioners are the same modulo compact operators.
Finally, we prove the generalized versions of the Korovkin type theorems in
[23]. Keywords: Completely positive maps, Frobenius norm, Pre-conditioners.
