A subspace V of a Banach space X is said to be complemented if there exists a (bounded) projection mapping X onto V . Obviously all subspaces of finite-dimension are complemented. The goal of this note is to show that there are (relatively) few monotonically complemented subspaces of finite-dimension in X = (C[a, b], ‖ · ‖∞); that is, finitedimensional subspaces V ⊂ X for which there exists a p...

Introduction A C∗-algebra is often thought of as the non-commutative generalization of a C(K)-space, i.e. the space of continuous functions on some locally compact Hausdorff space, vanishing at infinity. We go one step further, for we seek to compare the Banach space properties of C∗-algebras and their naturally complemented subspaces, with those of C(K)-spaces. (For a recent survey on C(K) spa...

The complemented subspace problem asks, in general, which closed subspaces M of a Banach space X are complemented; i.e. there exists a closed subspace N of X such that X = M ⊕ N? This problem is in the heart of the theory of Banach spaces and plays a key role in the development of the Banach space theory. Our aim is to investigate some new results on complemented subspaces, to present a history...