Embeddings into B-cyclic Banach spaces

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