We introduce a weaker notion of central subspace called almost central subspace, and we study Banach spaces that belong to the class (GC), introduced by Veselý (1997). In particular, we prove that if Y is an almost central subspace of a Banach space X such that Y is in the class (GC), then Y is a central subspace of X . We also prove that if Y is a semi M -ideal in a Banach space X such that Y is an almost central subspace of X, then Y is an M -ideal in X. Certain stability results for quotient spaces, injective tensor product spaces, and polyhedral direct sums of Banach spaces are also derived. © 2013 C. R. Jayanarayanan.