  In this paper we study conditions assuring that the Bishop-Phelps-Bollob\'as
property (BPBp, for short) is inherited by absolute summands of the range space
or of the domain space. Concretely, given a pair (X, Y) of Banach spaces having
the BPBp, (a) if Y1 is an absolute summand of Y, then (X, Y1) has the BPBp; (b)
if X1 is an absolute summand of X of type 1 or \infty, then (X1, Y) has the
BPBp. Besides, analogous results for the BPBp for compact operators and for the
density of norm attaining operators are also given. We also show that the
Bishop-Phelps-Bollob\'as property for numerical radius is inherited by absolute
summands of type 1 or \infty. Moreover, we provide analogous results for
numerical radius attaining operators and for the BPBp for numerical radius for
compact operators.
