Banach spaces of [formula omitted]-convergent sequences

Abstract

We study the space c0,I(X) of all bounded sequences (xn) in a Banach space X that I-converges to 0, endowed with the sup norm, where I is an ideal of subsets of N. Our results contribute to the development of a structural theory for these spaces. We show that two such spaces, c0,I and c0,J, are isometric exactly when the ideals I and J are isomorphic. Additionally, we analyze the connection of the well-known Katětov pre-order ≤K on ideals with some properties of the space c0,I. For instance, we show that I≤KJ exactly when there is a (not necessarily onto) Banach lattice isometry from c0,I to c0,J, satisfying some additional conditions.We present some lattice-theoretic properties of c0,I, particularly demonstrating that every closed ideal of ℓ∞ is equal to c0,I for some ideal I on N. We also show that certain classical Banach spaces are isometric to c0,I for some ideal I, such as the spaces ℓ∞(c0) and c0(ℓ∞). Finally, we provide several examples of ideals for which c0,I is not a Grothendieck space.