Embeddings into $𝔹$-Cyclic Banach Spaces

Abstract

Given a complete Boolean algebra $𝔹$ and nonzero $\pi\in 𝔹$, we introduce the notion of ${𝔹}_{\pi}$-embedding of a Banach space into a $𝔹$-cyclic Banach space by using lattice spaces as well as the notion of lattice ${𝔹}_{\pi}$-embedding of a Banach lattice into a $𝔹$-cyclic Banach lattice is also introduced. We also establish some criterion for the embedding of the space of continuous vector valued functions with values in an arbitrary Banach space into a $𝔹$-cyclic Banach space as well as a criterion for the lattice embedding of a space of continuous vector functions with values in an arbitrary Banach lattice into a $𝔹$-cyclic Banach lattice. These results allow us to outline some approach for isometric and isomorphic classification of $𝔹$-cyclic Banach spaces.