Embeddings into \(\mathbb{B}\)-Cyclic Banach Spaces

Tasoev, B. B.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 4.

Abstract:
For a complete Boolean algebra \(\mathbb{B}\) and nonzero \(\pi\in \mathbb{B}\), the notion of an \(\mathbb{B}_{\pi}\)-embedding of Banach spaces into \(\mathbb{B}\)-cyclic Banach spaces is introduced. The notion of a lattice  \(\mathbb{B}_{\pi}\)-embedding of Banach lattices into \(\mathbb{B}\)-cyclic Banach lattices is also introduced. A criterion for the \(\mathbb{B}_{\pi}\)-embedding of a space  of continuous vector-valued functions with values  in an arbitrary Banach space into a \(\mathbb{B}\)-cyclic Banach space is established, as well as a criterion for the lattice  \(\mathbb{B}_{\pi}\)-embedding of a space of continuous vector-valued functions with values  in an arbitrary Banach lattice into a \(\mathbb{B}\)-cyclic Banach lattice. The  obtained results allow us to outline an approach for isometric and isomorphic classification of  \(\mathbb{B}\)-cyclic Banach spaces. In the course of establishing the results, the tool of lattice-valued spaces was widely used.

Keywords: Banach lattice, \(\mathbb{B}\)-cyclic Banach space, isomorphic classification

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