Some theorems of Korovkin type

We take another approach to the well known theorem of Korovkin, in the following situation: X, Y are compact Hausdorff spaces, M is a unital subspace of the Banach space C(X) (respectively, C-R(X)) of all complex-valued (resp., real-valued) continuous functions on X, S subset of M a complex (resp., real) function space on X, {phi(n)} a sequence of unital linear contractions from M into C(Y) (resp., CR(Y)), and OD a linear isometry from M into C(Y) (resp., CR(Y)). We show, under the assumption that Pi(N) subset of Pi(T), where Pi(N) is the Choquet boundary for N = Span(boolean OR1less than or equal tonless than or equal toinfinity N-n), N-n = phi(n)(M) (n = 1, 2,..., infinity), and Pi(T) the Choquet boundary for T = phi(infinity)(S), that {phi(n)(f)} converges pointwise to phi(infinity)(f) for any f is an element of M provided {phi(n)(f)} converges pointwise to phi(infinity)(f) for any f is an element of S; that {phi(n)(f)} converges uniformly on any compact subset Of Pi(N) to phi(infinity)(f) for any f is an element of M provided {phi(n)(f)} converges uniformly to phi(infinity)(f) for any f is an element of S; and that, in the case where S is a function algebra, {phi(n)} norm converges to phi(infinity). on M provided {phi(n)(f)} norm converges to phi(infinity) on S. The proofs are in the spirit of the original one for the theorem of Korovkin.