We study the almost Daugavet property, a generalization of the Daugavet property. We analyze what kind of subspaces and sums of Banach spaces with the almost Daugavet property have this property as well. The main result of the paper is that if $ Z$ such that the quotient space $ X/Z$, then $ Z$ [ABSTRACT FROM AUTHOR]
Given a Boolean algebra A, we construct another Boolean algebra B with no uncountable well-ordered chains such that the Banach space of real-valued continuous functions C(KA) embeds isometrically into C(KB), where KA and KB are the Stone spaces of A and B, respectively. As a consequence we obtain the following: If there exists an isometrically universal Banach space for the class of Banach spaces of a given uncountable density κ, then there is another such space which is induced by a Boolean algebra which is not universal for Boolean algebras of cardinality κ. Such a phenomenon cannot happen on the level of separable Banach spaces and countable Boolean algebras. This is related to the open question of whether the existence of an isometrically universal Banach space and of a universal Boolean algebra are equivalent on the nonseparable level (both are true on the separable level). [ABSTRACT FROM AUTHOR]