Complemented subalgebras of the Baire-1 functions defined on the interval [0, 1]

An important problem in topology is to determine the effects on the function space of imposing some natural topological condition on the space X [5]. In this way, it is practical to characterize the complemented subalgebras of Banach algebras. In our investigation, we relate the second category property of [0,1] with certain properties of the complemented subalgebras of bounded real Baire-1 functions, β◦ 1([0,1]) (bounded functions of finite Baire index, 1 ([0,1])). We begin by recalling some definitions. Let A be a Banach algebra (resp., Banach space). Two subalgebras (resp., subspaces) M and N of A are complementary if A = M ⊕N . A projection on A is a continuous linear operator P : A→ A satisfying P2 = P. If M and N are complementary subalgebras (resp., subspaces) of A, then there exists a projection P on A such that the range of P is M (N). The norm (sup-norm) of the projection P is always equal to or greater than 1. Norm 1 projections play a crucial role in the study of complemented subalgebras (resp., subspaces) of Banach algebras (resp., Banach spaces) (see, e.g., [4]). If A is a finite-dimensional Banach space, then every nontrivial subspace of A is closed and complemented in A and does not contain a copy of A. This of course is more complicated for infinite-dimensional Banach spaces. Pelczynski has proved that every infinite-dimensional closed linear subspace of l1 contains a complemented subspace of l1 that is isomorphic to l1 [4, Theorem 6, page 74]. Also it has been proved that C(X), the ring of continuous functions on the compact topological space X is the direct sum of two proper subrings if and only if X is disconnected [5, Problem 1.B, page 20]. In this case, for every decomposition of C(X), there is an open compact partition {A,B} of X such that C(X) = C(A)⊕C(B). We want to establish a result similar to that of Pelczynski for the Banach algebra of bounded real Baire-1 functions defined on [0,1].