  Inspired by the recent work of Cascales et al., we introduce a generalized
concept of ACK structure on Banach spaces. Using this property, which we call
by the quasi-ACK structure, we are able to extend known universal properties on
range spaces concerning the density of norm attaining operators. We provide
sufficient conditions for quasi-ACK structure of spaces and results on the
stability of quasi-ACK structure. As a consequence, we present new examples
satisfying the (Lindenstrauss) property B$^k$, which have not been known
previously. We also prove that property B$^k$ is stable under injective tensor
products in certain cases. Moreover, ACK structure of some Banach spaces of
vector-valued holomorphic functions is also discussed, leading to new examples
of universal BPB range spaces for certain operator ideals.
