Geometry of Banach spaces having shrinking approximations of the identity

Let a, c greater than or equal to 0 and let B be a compact set of scalers. We introduce property (M) over dot * (a, B, c) of Banach spaces X which is a geometric property of Banach spaces generalizing property (M*) due to Kal ton: Using M* (a, B, c) with max \B\ + c > 1, we characterize intrinsically a large class of shrinking approximations of the identity, including those related to M-, u-, and h-ideals of compact operators. We also show that th e existence of these approximations of the identity is separably determined As an application, we study ideals of compact and approximable operators. In particular, this provides an alternative unified and easier approach to the theories of M-, u-, and h-ideals of compact operators. (C) Academie des Sciences/Elsevier, Paris.