An isometrically universal Banach space induced by a non-universal Boolean algebra

Abstract

Abstract. 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(K A ) embeds isometrically into C(K B ), where K A and K B 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 if 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).

On disconnected images of connected spaces

“…With this kind of notation, we can say that a family C of pseudoclopens is a ⋆ chain if * P (a 1 , a 2 ) for any distinct a 1 .a 2 ∈ C. Our work has been motivated by the following result of Brech and Koszmider, which was used in [2] as an instrumental tool to provide some examples in the theory of Banach spaces of continuous functions.…”

“…Our work has been motivated by the following result of Brech and Koszmider, which was used in [2] as an instrumental tool to provide some examples in the theory of Banach spaces of continuous functions.…”